But after leaving it for about two weeks, it no longer started at all. So I've dug deep into the engine for the first time ever. I checked the fuel filter, which is pretty new and pretty clean, and the spark plugs all look pretty good. I pulled off the manifold and poured in some carb cleaner, then tried pouring gasoline directly into the carb. It DOES try to start after that.
So, I went ahead and ordered a VW replacement fuel pump. For $30 it seems like I might as well replace it, since I know nothing about what shape the old one is in and it seemed like a likely culprit. I replaced that tonight, with a couple potential issues: the one I got is a somewhat different style than the one that was in there. It looks like this:

Whereas the old one has the two line ports side-by-side and facing forward. But I figured it's here and I might as well give it a shot. Since the lines now have to run to the rear, rather than the front, I took the opportunity to shorten the line from the filter by bypassing the Y to the heater altogether... I remember reading that the heater itself is sometimes a problem and probably ought to be gotten rid of regardless. So it's out of the loop now.
And, well... it almost sorta works? It still wouldn't start initially, even after a good 30 seconds of cranking. So I once again gave the carb a little shot of juice, and that combined with a little gas was actually enough to get it into an idle... but a pretty rough one that died a few times. In addition to belching smoke out of both carbs at one point and backfiring super sharply on two other occasions, which is fun. That's when I decided to stop before the neighbors come looking for me.
So... where am I at now? I've still got the air filter and the manifold off... do I just need to get those back on and give it some time? Is that pump just not gonna cut it? Or is there something still fouled up I need to be hunting for?
When replying, remember to respond as if speaking to a particularly slow child with basically zero automotive experience. Slightly helpfully the only other car I've ever worked on was a 1968 Karmann Ghia, but I'm still essentially an idiot.