Haven't read the WHOLE thread, but a few points.
DISCLAIMER: This a hyperbolic, drunk, and probably skewed version of Middle Earth history and facts.
In LOTR, most of the flashy magic comes from imbuing power into things. I.E. Ring, Swords, Maces, Whips, et al. (sometimes people, but that generally turns them into Orcs and/or Balrogs--afer a whole bunch of torture and suffering ofc) and generally, in the Third Age of Middle Earth, the only beings who have access to the power to make/wield these things are evil as heck. In the Silmarillion this shit is widespread. You have Elven and Dwarven smiths measuring peens over who can create the coolest and magic-est items. Dwarven armor is so good in that age, Dragonfire can't melt it. There is a Tiara of ultimate power (leading to pretty much everything that fucks over the world). After a while, considering how destructive and easy-to-acquire all the blatantly magic stuff is the Gods start to think about gun control and basically take all the cool people aside and say "Hey listen, we appreciate your baller weapons and items of power, but uh... it's starting to kind of mess up this awesome world."
[insert tales of Morgoth, a bunch of evil biz, and the forging of the rings of power]
Later, Gandalf is actually a pretty beast guy. If he wanted to, I have no doubt he could have just fireballed Denethor in the face and taken control of Gondor himself, BUT, not only was he under orders from the Godlier Gods (I mean, technically, Gandalf is the ME equivalent of an angel or minor god, right?) not to show his full power to the peoples of Middle Earth, and thanks to good ole Saruman, the average Gondorian mistrusted and hated wizards, so I doubt Gandalf doing anything flashy would have helped. You might ask, but what about when he was with the fellowship or Thorin's party? Well first of all Dwarves are even more distrustful of magic than uneducated peasants. Secondly, he's still under orders, and Gandalf pretty much takes everything to an extreme. Finally, I'm certain he would have attracted too much attention to either party if he just went full wizard and start turning people into red mist or charred corpses.
So that is that.
But as a couple of people pointed out, in gaming systems, whether it's P&P or video games, Wizards basically suck dick at low levels, and to make up for it, they become gods when they start maxing out. Also, it's probably not super fun, in a gaming context, to do what Gandalf did, which is basically just snoop around and subtly influence events for millennia.
I know we're using fireballs as an example, but it's really all overt magic, right? There was a time when Wizards were universally viewed with fear and suspicion. Mostly because of the uncertainty of their character. What COULD they do? Is their magic subtle and unseen? Are they slowly driving us insane? I think that stuff is pretty difficult quantize for gaming systems, and while less-difficult to portray on screen (given many famous wizards with highly political personalities), it just doesn't provide a lot of visual evidence. It's just easier to show a huge fireball or whatever and go WOW. That dude can do magic.
tl;dr the demands of easily-understood visual appearance are what led to the modern concept of the wizard.