I just fixed the problem and now it spins up without the transistors overheating, but the method that I used was hilariously stupid. I figured that it just needed to be spun for a while to loosen things up, so I rigged up a contraption using my dad's drill press, a hole saw, and a roll of electrical tape (don't worry, the electrical tape, not the hole saw, was the thing that actually touched the motor) that would spin the drive for me. I let it run for a few hours and the spindle motor felt a lot easier to spin afterwards. I tried powering it up and it spun up much quicker than with the hair dryer and now the transistors are running much cooler. It's really loud, so I'm going to let it run upside down for a few hours to redistribute the grease and quiet it down and then I'll reconnect the analog and controller boards and see what happens. After all of this work getting the spindle motor to spin, I sure hope that the drive is good!