The power supply in the beige G3 minitower, like that used in the other G3/G4 tower Macs until the Gigabit models (which had a 28v line for the new ADC monitors), is basically a standard ATX unit but with slightly longer cables. It's real easy to install a higher-wattage ATX unit in one of those.
Because the beige G3 MT's Gossamer logic board was designed for use in multiple enclosures with multiple power supplies, there is a jumper on the bottom rear of the logic board that says MAC SUPPLY/PC SUPPLY. If your power supply is from an ATX unit or a MT, the jumper is on the PC SUPPLY side. If you have a DT or AIO, it's on the MAC SUPPLY position.
I had a G3 MT do a similar thing where it powers up but won't boot. I only got it to function properly by doing crazy combinations of removing components and then trying to power up.
I tried:
With ROM
With no ROM
With RAM in various configurations
With no RAM
With processor (original or upgraded)
Without processor
Changing the system bus speed to a different value (66, 75, or 83MHz)
Changing the processor speed jumpers (lower and higher than original)
With each iteration I pressed the CUDA reset button and there was not a PRAM battery installed.
Eventually the system decided to function again and I tweaked it to run an 83MHz bus, 375Mhz processor, 384MB RAM, Voodoo3, combo USB/FW card, and a fast SCSI card. I don't know what ultimately made it work but I put it all back together and it has been pretty reliable since.