I've managed to fix the issue.
After inspecting the card in detail, replacing the missing caps and measuring all the other components on the board and still not finding any issues, I assumed that the issue had to be related to the damaged via on the rear.
I followed the track from the damaged via to another via which connects to one of the pins of a ram chip on the rear. Testing continuity from that via to the broken via always gave me a connection but clearly the track still continues on the front and I needed to test further to be sure. (Maybe the slight pressure from the tip of my multimeter was enough to connect a break)
To continue tracing the connection on the front of the board, I put a small piece of wire through the via and traced it to a pin on the nearest TMU. Then I checked continuity all the way from the ram chip on the rear to the pin on the TMU in the front and sure enough, no connectivity.
I put the card under the microscope and scratched at the trace of the broken via on the rear and found that there was a tiny piece of trace that was lifted from the board. Scratching at it caused that piece to break off and leave a gap between the via and the trace that leads to the ram chip.
I put a fine piece of copper wire into the via and soldered it to the trace.
After that, I tested continuity from the ram chip to the TMU again and this time, there was connectivity.
Tested the card in my PC again and finally the texture errors were gone and everything worked as expected.
I don't have much experience with electronics repairs so this is all just a learning experience for me. Luckily it worked out this time.