Firewire is a great high speed standard of the day compared to USB 1.1; very much like an enhanced version of SCSI but you can see why it was left behind. I've a couple of FW devices that play nicely in later Macs but not all, power and signaling issues are also a consideration.
Nice, I’d suggest practicing on another old PCB first, as both are probably pretty easy to resurrect with some standard recapping/troubleshooting if clean.
I don’t know what that black adhesive but not normally placed. Just make sure it’s not conductive and once you know the Mac works it might come off/dissolve with IPA or Acetone. Or just leave it.
Also recently picked up a couple of 7300 machines; a 7300/180 and 7300/200; both exhibited the same behavior, no chime bar some static on the speaker. PSUs checked out fine. For both, it was the CPU daughter card and a stick of bad RAM in each. Amusingly switching the CPU daughter card on...
Is the CRT good - have you tried a nubus video card on it? The 840AV has 1MB soldered VRAM I’d pull whatever is in the 4 X VRAM slots as first troubleshooting step.
https://www.instructables.com/Imac-G5-DIY-capacitors-repair/?amp_page=true
Some starters here, you’ll need to research tools and capacitor ratings next.
Welcome - quick answer: no, because iMacs of this era are plagued by bad caps, even if just a few appear leaking you can bet many others are marginal. If you don't have the materials I'd look out for another iMac G5 which is cheap and has nil cap leakage - for now.
It’ll be the RGB wires on the yoke going to the analogue board, resolder all these with fresh solder and flux. Also clean the 575 edge connector thoroughly.