This is turning into a pleasant walk down Memory Lane!
Have you tried powering on using the button on the computer case?
Not sure about the Beige AIO (nicknamed "The Molar" because it reminded someone of a great tooth) but the mini-towers had a rather flimsy mounting system for the case-mounted power button. After some use, the plastic mounts that held part of the switch assembly would flex too much when the user depressed the button. The result was the contact did not remain closed long enough to get a proper start.
The button assembly on the mini-tower was easy enough to access and you could add support to overcome the problem. However, this issue did not prevent the computer starting from the power switch on the keyboard, thus raising the the ADB keyboards questions Jan and I posted.
Have you removed any PCI cards that came installed? If not remove any in the three PCI slots but leave the card in the longer inboard slot (more below).
Tried a CUDA reset? Press and hold the CUDA rest button for 20 seconds. I'm not sure that this will help as the old battery was surely long ago drained but, if the dying battery corrupted the contents of PRAM, the CUDA reset clears it all.
Is the Personality card is properly seated? They work loose with age and vibration and can cause a failure to start. It lives in the PERCH slot inboard of the three PCI slots. Remove its retaining screw and completely remove the card. Clean its contacts and blow out the slot. Reinsert, then remove and insert again to freshen the contacts. This card must be level across its entire length and it take some force to accomplish that.
I remembered one bug in that series: the voltage regulator, located near the PRAM battery cage. One brand, "Royal," was clearly not up the the job and its failure could give the symptoms you are seeing. The brand is clearly marked. If Royal is not clearly visiable, then the regulator is not the problem
I don't know if you have one but here is a logic board diagram, from the MacGurus website:
http://www.macgurus.com/products/motherboards/mbppcg3allone.php
There is also some lovely information contemporary with your computer that is useful.