Thanks for all the help everyone -- this has been extremely helpful!
I've thrown in the towel on repairing the power supply. I just don't know enough about what I'm doing, as is clear. While the sparks and smoke were fun and all with the case on, not so much with the case off
My current plan is to get a new linear regulated power supply such as this guy:
And to use a 80-100W bulb that uses the same mount (there are many) and is matched to the voltage of whatever power supply I find. In some cases, I'm even seeing power supplies where I can adjust the voltage so maybe if I find one of those cheaply enough I won't even have to change the bulb, and could switch to a 12V later without getting a new supply.
It seems that most voltage regulated supplies are switched nowadays, so it has been trickier than I expected to find a linear regulated supply. I have a few options I'm working on, but they all are of the hobby type. While I'm fine with that, any other suggestions on a good source for a new linear regulated supply?