I gave it another shot today, but since I couldn't really visually find any bad connections, this time I armed myself with a new tool - cheap Aliexpress logic probe,
with the intent of poking at all of the EPROM pins and seeing if I can see something off.
I found two suspicious things - pin 15 and 17 on 2 topmost graphics chips either didn't seem to display any activity, or only very occasionally the "pulse" diode lit up, while every other pin displayed some kind of logic state every time. I couldn't figure out what's the deal with pin 15, it doesn't seem like any traces go to it (so maybe it is supposed to be floating), but I actually did find what looked like broken connection from pin 17 to the via not far above it (at least I think that's how this connection is supposed to look like, hard to tell with socket in the way). I quickly soldered a jumper and booted up the board again, and....
... absolutely nothing changed, vertical bars were still there. And besides, the pin activity didn't change at all when I tried to push the board (which normally fixes the vertical bars), so maybe I should have expected it
So, back to the drawing board. For what felt like the millionth time, I looked closely at the board with magnifying glass, and this time what caught my attention was the graphics chip (TC0630FDP), the very corner closest to camera:
The pads behind the pins don't look the best - but in fact, it was way worse when I got this board, it was covered with brown gunk, especially in that area (you can still see it on the label) - so the first thing I did before even starting the conversion was thoroughly cleaning the board (had to use soapy water, isopropyl alcohol didn't cut it). I thought this area was worth examining since this is the graphics chip, and it's not too far from the spot on the board that seems most sensitive to pushing. Enhance:
That's kind of hard to see on the photo, but it looks like the pad of the last pin (close to number 157) has way less solder than other pads - it doesn't look like it actually connects with the pin. I poked the pin very carefully... and sure enough, it moved very easily. I booted the board again and now simply touched the corner of the chip very lightly - the vertical bars easily disappear. At this point I'm convinced this is the culprit!
So I tried my best to solder it in place, but the pads being in sorry state made it a bit difficult (I wasn't able to clean it up much better than this), but eventually I managed for the solder to stick and the pin now didn't move:
It still looks it's floating on this photo, but it's in fact firmly attached (I didn't manage to make it look better than this, the pads really don't take solder very well at this point), and multimeter test convinced me that the connection is now reliable enough. But does it work?
Yes it does! No vertical bars, no magic finger necessary. If this didn't work I was planning to connect enamel wire to the pin and solder it to the trace, but at this point I don't want to mess with these tiny pins anymore.
So suprisingly, it turned out that the problem was nowhere near the pins of the mask ROMs that I had to desolder - I either dislodged this pin while cleaning this area of the board, or it was already dislodged when I got it and it simply didn't visibly manifest when running Taito Cup Finals and Darius Gaiden just made the issue noticeable. Who knows

Time to enjoy one of my favourite shmups of all time on the original hardware, and I couldn't achieve this without this guide so many thanks yet again!