It's a risky move for sure but I'm not convinced his dad will launch a military campaign just because his son took a sword that was his by birthright. I imagine there will be some conflict though.
The Lannisters sent a whole military expedition to Valyria to retrieve their lost Valyrian sword, no survivors returned. Randyll Tarly will also not need to launch a military campaign, as he is a General of the Reach's military, and Sam is currently in the Reach and his destination still in the Reach - which his father knows where he's going. For 500 years that sword has been in his family and Tarly will not let this slide.
This was a very poor life choice on Sam's part. Sam's usually not this stupid too, so it remains to see how this unfolds. Would be rather anticlimactic to have Sam be killed over this sword and the Wall get bumrushed by an enemy that cannot be stopped with conventional warfare. It's ok though, Bran will save them.
If I wanted to create a new weapon against the White Walkers I would definitely start by examining existing weapons that work. Maybe it only needs to be 1% Valyrian to work or maybe you just need a tip, so you could reforge that one largely ornamental blade into hundreds of effective weapons.
Sam is a scholar, not an alchemist - they already had obsidian and valyrian steel at the Wall, what they didn't have was access to the great archives of Oldtown which is basically their library of Alexandria. In short, if the kryptonite seckrit is not there, it's lost to time. Think back to season 1 where Maester Lewin (Luwin? Fucking GRRM spellings ;
D), shows Bran his valyrian steel chainlink, showing he has studied magic. He also adds that the maesters of Oldtown now consider that link to be rather useless study of myths and superstition, but all the same they are the only people in the world besides maybe the alchemists who have (and the maesters have the largest) collection of the oldest myths, scrolls, texts and so on - with scholars dedicated to their study. If studying Valyrian steel with Sam's skillset would yield results, then he would take Longclaw or any of their obsidian weapons from Jon. When Stannis and Sam talk in Season 5 they talk of a few things (including Sam's father), but most importantly they talk of white walkers, specifically killing them. Stannis asks Sam how he killed the white walkers, Sam says with dragonglass. Stannis, having lived on Dragonstone (a whole island made out of dragonglass) is the closest thing to an obsidian expert they have at the wall and asks Sam why dragonglass killed the white walker.
Sam says he doesn't know why it works, only that it does - and that in the manuscripts he studied, the children of the forest were said to have hunted with dragonglass weapons.
To quote:
Sam: 'I've seen it your grace.'
Stannis: 'Seen what (m8)'
Sam: 'The army of the dead. And when they come-'
Stannis: '-We have to know how to fight them. Keep reading Samwell Tarly.'
From Season 5 episode 5. Also quite adorable is that in this scene, you can see Sam feeling self conscious about not being a soldier like his father - but how he looks after Stannis validates his value as a reader is inspiring. You can see it means a lot to Sam, to have this King who is not his friend validate him for his talent where his father would not. Story telling!
I digress, because after Sam realizes his value is in finding out how to kill the White Walkers, he reads through all the archives of Castle Black and runs out of material to read - at which point he goes to Jon and says he needs to go to Oldtown to access their archives and find out everything they know of the white walkers and the children of the forest. As it stands they don't know what makes Valyrian steel or Obsidian so special, they don't know how the White Walkers control winter or the dead, how they will interact with the Wall, how they reproduce, how they communicate e.t.c. - every single man at the wall could have a Valyrian steel greatsword, and not knowing how to fight the White Walkers would mean nothing, they will lose. This is why Jon ultimately authorizes Sam's journey south - they need to know their enemy.
Take your idea for example - put yourself in Sam's shoes. He's already examined all of their weapons that have been proven to kill white walkers, namely Valyrian steel and dragonglass. Let's assume Sam's also starting to believe that the Valyrians had real magic and the Children of the Forest and their magic were both real (after having seen the armies of the dead walk, anything is possible). He doesn't know if dragonglass sourced from Dragonstone will have the same effect as the Dragonglass from the Fist of the First Men, enchanted by the COTF. He doesn't know if melting and reforging Longclaw or even Heartsbane into a hundred arrowheads will not result in breaking whatever magic the Valryians have cast on it, a conundrum made even worse since the COTF just became extinct this season and the method of producing Valyrian steel has been lost for a looooong time, and the people who have the knowledge to reforge Valyrian steel are a prized minority - it took Tywin Lannister, the most powerful man in Westeros to reforge Ice into Widow's Bane and Oathkeeper. Even then he had to get his blacksmith from far away in Essos, being perhaps one of three or so people alive still capable of reforging Valyrian steel.
The wall is 300 miles wide by source standards, 500 miles by Sam's show estimates and only 3 of the 19 castles are manned. Both the wildlings that made it through the wall or out of hardhome are greatly worn down, with most of their fighters having died against the Night's Watch, Stannis or the White Walkers. The Night's Watch at this point is a skeleton crew. Stannis's men are dead or deserted. The White Walkers numbers are unknown, as all rangings have ceased due to the Night's Watch having no manpower to do so, the gate having been frozen shut and the last ranging resulting in their force being eaten by dead people or mucked about by legends from fookin Jin Alley. Of the powers known to the Night's Watch and the Wildlings at the wall, seeming invincibility to anything but Valyrian steel and dragonglass, assuming this weakness is a universal property to the material and not the enchantment - with the enchantment being specific to the White Walker and not the material (so white walkers are weak to dragonglass because dragonglass was used to make the first white walker, not that the daggers found are uniquely enchanted. Valyrian steel is the opposite, being straight up magic - never needing sharpening, cutting through most other metals and fucking up white walkers like it's NHS budget).
They don't know why or the specifics, those answers being in Oldtown. They know the dead have a tendency to not stay dead for long unless you burn them, but they don't know why this happens - for example the body they brought to the south of the wall coming to life on its own, the Wildlings wanting to burn Jon's corpse, and the Night King raising the casualties at hardhome. No one at the Wall has seen enough or understands anything about the process of reanimation, only that you should burn all corpses just in case. Jon has witnessed the White Walkers not realizing that they are vulnerable to Valyrian steel - it comes as a shock when Jon kills the White Walker where the Thenn Chieftain's greataxe shatters, but now they've learned this weakness about themself too. They know the White Walkers have a seeming ability to control the weather itself, and they know Winter is coming - if there is a connection they don't know for sure, and they don't know whether when winter comes and all the people who have died of starvation or cold in the south or have died from battle... Are going to stay dead. They also know that the wights are especially weak to fire but otherwise cannot be killed like regular people, but the White Walkers seem to shrug off fire like it's nothing. They also don't have wildfire to play with, but an educated guess is since wildfire is magic greek fire, it would probably burninate the shit out of them. Shame the Night's Watch has no knowledge of Wildfire - Maybe Davos may inform them? Possible. No certainties, because that knowledge is in Oldtown.
The fable about ice spiders is probably going to be just a fable because CGI budget yo, but the rest is confirmed to the people at the wall.
Then there's the issue of the Wall itself and how it interacts with the White Walkers. As it stands the Night's Watch could not hold off the Wildlings, logically this raises the question of why the White Walkers haven't already attacked the Wall with their army of the great dead and won already. All they know is that the White Walkers have been marching south since season 1, have been systematically exterminating and reanimating the wildlings and night's watch people north of the wall - and that the Wall was originally built not to keep the wildlings out, but was built by the first men and the COTF to keep the White Walkers out. They need to know what mechanisms of magic dickery are available in the Wall to keep the White Walkers out or they're stuck trying to fight conventionally against an enemy that cannot be defeated in conventional combat with forces so understrength they couldn't even hold off a conventional enemy.
So it seems very out of character for Sam, having just discovered his true calling as a scholar and not a soldier - to jeopardize the safety of the world in order to play soldier and risk completely undoing his own well-thought out plan. This does seem like a decision made to heighten the stakes since otherwise Sam would be reasonable and have just secured a safe home and noble education for his family, but it is one quite out of character for Sam and the only way this will make sense if Randyl Tarly similarly acts out of character.
Rather interesting here is that there are two people who are studying the weaknesses of the White Walkers. Well, only one is, the other is faffing about with Daddy issues and showing off in front of his girlfriend. Whilst Sam goes south to Oldtown, Bran went north. Sam has a conversation with Jojen, the greenseer Reed. Jojen begins telling Sam that even South of the Wall is not safe from the White Walkers, and the Night's Watch can't stop them and not even all the Kings of Westeros and their armies combined can stop them. Bran has to go north to find out how to defeat the White Walkers - and Sam gives Bran's crew the dragonglass he found at the Fist of the First Men, and Bran actually begins researching (well, looking into the past) to see the creation of the White Walkers with an obsidian dagger, meaning he's on the road to discovering how to defeat the White Walkers where no one else can. Well, except for maybe Sam - if he doesn't get himself killed. Sam even gives this great speech about how he got lucky with one White Walker, and there were many White Walkers - and for each of them, too many dead soldiers to count.
In a beautiful way Sam is doing what Bran is, looking into the past - reading the old histories of Westeros, seeing the past without Bran's gift. Or rather, he would be if he wasn't faffing about.
To conclude this Heartsbane is not an ornamental sword, for it is the greatsword Randyll Tarly uses in battle. If they managed to get one of the few people alive capable of reforging Valyrian steel to do the job for free and didn't all get killed by Randyll Tarly, they would still be horribly incapable of stopping the White Walkers - and this is all knowledge Sam knows, being one of the only two people alive researching how to stop the White Walkers.
Tl;dr: Sam spent 5 seasons learning that the pen is mightier than the sword, yet when confronted with the sword or the book, chooses the sword.