Unfortunately, I don't think the Amiga-to-VGA adapter is going to solve your problems unless it's the so-called
Commodore silverbox adapter (or compatible - there are a few) and your monitor is capable of 15KHz horizontal. The straight-through 23-pin to 15-pin converters don't work on A2000s (or at least, not on mine) and will prevent the machine from booting - something to do with the machine's genlock features. The silverbox contains some extra circuitry to deal with this issue. The straight-through pin adapters do work on later Amigas.
Actually, speaking of genlock issues, have you been running all your tests with the A520 still connected? Just a thought, but try booting up without it and see if it makes a difference. It shouldn't, but if you've tried every other variable...
After that, I think it's time to start playing with the 2320's potentiometer per the manual. While CRTs used to display garbage if something wasn't calibrated correctly, LCDs tend to just give up and not display anything.