Alch's biggest advantage is his ult, most definitely. 1.7 BAT to 1.0 BAT is absurdly strong, especially combined with the 750 hp (39 str of hp) and 60 ms the ult gives him. The BAT decrease is stupidly strong; it's very similar to troll ult in effect when he has an ac and shadowblade. The irony of alchemist is that his ult's BAT allows the hero with the strongest farming mechanic to actually need less farm to achieve the same DPS as other heroes. And he's a safer slot 1 than any other, because he has a stun and zoning skills.
IMO, you can fix bat by decreasing firefly duration or increases the cd. Napalm is strong, lasso is strong, flamebreak is decent, but he has 18 seconds of high damage terrain walk that allows him to scout, escape, and engage very easily. Other heroes have blinks, jumps, even spectral dagger, but none of those allow the user to move anywhere he wants for 45% of the time. It makes him insanely hard to gank early: if he sees you, he pops firefly and walks away, over any terrain. If you can keep up with him, you end up fighting on the fire, which procs napalm stacks on you. Harder and more dangerous to engage onto than tinker. 70% turn rate slow is just icing.
Void, sniper, and bara all have bashes at equal or greater chances (that use prd) than basher that deal damage in addition to the bash effect. Basher might be too cheap for what it is, but I don't think you can call it the problem when most heroes cannot abuse it like am and alch. The key is a low base attack time that allows a relatively small amount of ias to get you a lot more attacks per second. People get it on naix as well, but he has a +80 ias on rank 4 rage every 17 seconds.
Magic immunity is the cat that was in the bag, so to speak. BKB is requisite on pretty much every carry, because there are so many potent skills that don't pierce magic immunity. However, because BKB counters the vast majority of skills, it means that supports and nukers are only effective until the bkb's start appearing. It's relatively rare to see pipe, because people who need to live through the fight buy bkb and bkb is relatively cheap.
Ideas to extend the influence of nukers and aggressive supports:
add a white king bar that uses a single ogre club and a 1500 gold recipe. This item makes the user immune to magical stuns, hexes, and attack slows, but does not stop movement slows, physical spells, damage, or silences. It does not have scaling duration or cd and can be disassembled.
Increase the recipe cost of bkb to 2600. BKB now costs 5200 to build; much more in-line with its influence in fights.
Decrease the recipe cost of pipe by 600 gold.
With WKB, there's a cheap response to stun-lock roaming, but also leaves the owner more vulnerable to effects that are more or less useless right now. So, indirect buffs to veno, aa, silencer, skywrath, slardar, riki, viper and more when combined with the relative inaccessibility of bkb. Might be an indirect buff to heroes with regular magic immunity skills, like omni, naix, and jug, and str carries in general.
BKB turns into an actual late game item, now that it doesn't cost 900 more than a freaking shadow blade. This, as well, might be a significant indirect buff to the ridiculous farmers in the game right now, alch and am.
The price point on WKB and pipe now mean they can actually be picked up by slot 4 or 5, perhaps allowing them get off spells during the late game without standing 1000 units behind their carries. Because it's so much harder for carries to be totally immune to all magic damage at 20 minutes, pipe is suddenly a lot more important for winning teamfights. Decreased price also makes push strats viable again (against more than just ratdota; how about that tower score).
Almost certainly terribly flawed and full of second order effects I haven't thought out in the 10 minutes I took to come up with the ideas. However, if you can make BKB a situational item, you've already drastically changed the shape of dota.