Even that may not be a hard requirement. Tiger 10.4.11 boots natively on my Wallstreet 292 MHz, and it has neither FW nor AGP. It does require XPostfacto to install, but not to boot.
What happens to flyback transformers in many cases is over the years heat expansion and contraction cracks the solder joints on the PC board. It's a good idea if you're handy with a soldering iron to touch up the solder on hot parts on the analog board of any old Mac which comes under your care. That will prevent a lot of failures. While you're looking around, look for leaky or puffed up electrolytic capacitors. The usual safety warnings and precautions apply, of course. There are lethal voltages around CRT circuits and you could kill yourself or get a nasty shock. 8-o