This was solved by the first reply - turney power switch on the back was set to keep it on. I followed traces all over the board and hopefully I would have found that power switch at the back sooner rather than later, but thank you for saving me time. Good luck to everyone else with their similarish problems.
You can always turn the button on the PSU by 90º to see if it was in the "always on" position. This was done as a failsafe on these machines in case they were used as servers and power was interrupted.
Of course, it could just be a wonky PSU or even something on the board. I have a IIci with the exact same issue, although the power-up isn't immediate. The board was re-capped and it's been problematic since, which I think is just the result of cursory work.
A few of us have come across the same problem recently - LCD stays on black once both logic board and LCD controller have been recapped, and it appears to be an unknown fault on the PB100 logic board itself. You can interchange the "black" LCD to another PB100 board and it works.
What that fault is I'm stumped, but more of us would like to know!