Hmm, did some trials and it seems that the computer starts up (almost) every time if I plug in the battery first. It doesn’t boot up with charger only (I guess this is expected) and I need to RESET + NMI it to get it to boot if I plug in the battery after the charger. So... most probably it works from power mgmt perspective and the sad Mac is because of a bad trace etc.
Well, after years of putting it off I finally opened up the PDQ. It wasn't easy, but easier than opening up an iBook. I removed the PRAM battery, but I did not replace it with a new one. While I do like the PDQ I don't use it as often as I do my other PPC Macs. I only put PRAM batteries in Macs that I use frequently. I used a stopwatch to time the delay and the result was about 40 seconds under OS 8, both with and without the PRAM battery, so I was off with my earlier estimates. The PRAM battery wasn't leaking, thankfully, but better be safe than sorry. I should also mention that I don't use a main battery with this laptop. The ones I have don't work. I reduced the RAM from 256 MB to 128 MB and that brought the delay down to about 30 seconds. I decided to just live with the delay.