You may be able to locate a replacement logic board that has been repaired or refurbished;
or perhaps a used board that has been tested and determined to be acceptably functional.
In some regions, where a light industrial base exists, some specialist companies can do a
major refurbishment of a logic board and perhaps graphic cards, for different Mac models.
These companies have workstations that are fairly modern and they can perform micro-
soldering and accurate replacement of the defective tiny bits on the logic boards, anew.
One of several companies that can do specialized repair to logic boards (or refurbish a
complete computer; which sometimes is better because the entire unit can be tested)
is in US. Shipping can be a problem; even for bare logic boards: wegenermedia.com
You may be able to acquire a pre-repaired or used pulled working parts from a vendor online.
https://duckduckgo.com/?q=refurbished+logic+board+powerpc+imac+G4&t=ffsb
Some potential sources include powerbookmedic, powermax, and a few dozen who are mac
centric resellers; some are full repair services in addition to parts depot repository sources.
Sorry that your vintage obsolete G4 iMac model has issues. I own a few with similar needs.
A few times a year, I see low priced used iMac G4s that I wonder about, but I don't bite.
In any event...
Good luck & happy computing! 🙂