What AC adapter are you using? Is it a stock adapter, or a cheapo third party one? If the latter, that might be the culprit. The commercially available third party adapters on the market specially made for old consoles are almost all garbage that will damage your console due to not supplying enough current.
The next step I'd take is visually examining the electrolytic capacitors on the board. Those are generally the most common failure point for 20-30 year old+ electronics. Look for any signs of leakage, corrosion, or buldging. You might have washed some of this away by cleaning it first, so if it is there it might be hiding. If I recall some models of the SNES use SMD electrolytic capacitors.
Don't spend money on cables unless you want to upgrade to s-video.
As far as resources for troubleshooting, I suggest you find out what board revision you have and give this page a thorough look over:
https://console5.com/wiki/SNES...
HQ board pics would help. Give some close ups of the caps in the A/V area, and power supply area if you can.
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