It’s where I’m inclined to guess (as opposed to the logic board). Were this an iMac G5 (from the same general time period), the culprit with the system not powering on (or staying on) would be the notorious liquid-filled capacitors which failed in these (note: many liquid-filled capacitors will work fine for a long time, but there was a quality control problem with said capacitors in the mid-aughts, and Apple weren’t the only manufacturer to deal with it).
The capacitors in the PowerBook, due to size, use solid, tantalum-based capacitors. Even so, the symptoms you’re having could either be that or, as
@eyoungren hypothesized, the PRAM battery in yours is just completely dead and needs to be replaced (which is probably a good idea to do anyhow). I don’t think your logic board, which is what counts the most, is bad, but I do think something involving power delivery and/or power storage (like a capacitor or ancillary battery which functions as
kind of a “super-capacitor”) is amiss.
If you ultimately try to pick up a replacement PRAM battery for it, this may be easier to find online (though don’t expect finding a new-old-stock one). In the longer run, pulling out that HDD and replacing it with an SSD might also be advised. This is because the PATA/IDE HDD has two sets of power rails — one, for the internal logic, and the other for supplying power to the moving parts; pulling a lot of power from especially the latter (not surprising, really) might be too much for the other power components to handle, especially if their power draw exceeds what the capacitors can hold “in the tank”, so to speak. An older HDD may, especially if one not used for a long time, have moving parts which are resisting movement relative to what they once allowed when still newl; resistance to movement requires more power to overcome that resistance, which here might be too big an ask for your PowerBook.
That’s my working hypothesis. You’ll need to set up a short shopping list to find out: replace the PRAM battery; replace HDD with an SSD (like an IDE-to-m.2-SATA adapter, plus SSD blade); and, if necessary, look out for a used, working DC-in board.