So, after inse's reply and looking at some reference designs, I decided to check the KA7552 power supply.
While measuring the voltage, I checked again the outputs and realised the PSU had started working again... weird.
I checked the KA7552 supply capacitor, C6, having a nominal value of 22uF. It measured 15uF with my UT210T (a bit suspicious, but I wasn't sure how well it would measure while still on the pcb) and 5uF with my BM235 (now very suspicious).
I removed it and found a bit of leakage on one of its legs. So, I replaced it with a new one.
That was definitely an issue. Good guess Miti!
Unfortunately, I have another problem now. The display is glitching a bit.
I don't know if it's related or not.
I measured the AC component on the outputs from the power supply and there's virtually none (a few millivolts).
I don't have a second scope to properly check for glitches.
I disconnected the backlight to check if it was related to the high voltage and it made no difference.
I found this video of a similar problem:
Mine it not this bad, but I'm seeing the same black horizontal bands.
I checked the electrolytic caps accessible on the back of the LCD module without taking it apart (presumably the ones the guy in the video had to replace) and they measure 3.5uF, which appears to be okay (text on them is "3.3 35V").
I also tried to heat up the LCD module and PSU a bit with a hairdryer and it didn't seem to make a difference, though I did not want to use too much heat.
I'm undecided if I should replace those caps on the LCD module, the remaining caps on the PSU or look for another scope to borrow to dig deeper...