Well, I was wrong. It only worked before because it was warmed up. The next day, it was back to several retries before it would start.
I googled around and saw on Dave's channel and realized that there were two boards- an SMD and a through-hole, whereas my monitor only had an SMD board. Then it hit me... the power supply is separate.
I soldered some leads on the monitor and hooked up my power supply and everything worked fine. So I sliced open both the wall wart and my finger to find a bulged cap in there as well. I replaced that cap ("Su'scon" brand, 105c, 25v, 680mF- 10.3mm x 17mm) with a similarly rated one I had handy (chongx, 105c, 50v, 470mF), and it was just the right size to fit back in the wall wart for reassembly. Everything seems to function perfectly well and silently now.
Would there be any issue with this cap being a bit lower capacitance?