I was cleaning a bit, and I found my stash of Gamecube games, so I decided to buy an HDMI adapter for my old Gamecube, and it arrived today. This is the one I bought: https://www.amazon.com/dp/B09TR1LM9H
I've tried a couple of games so far, and the results are more positive than I expected. It's obviously not as clear a picture as a modern console, but it's good enough. I'm looking forward to playing Metroid Prime 2, which I was surprised to find was still factory sealed.
The clock battery probably needs replacing, as the system time was very close to midnight on March 4, 2019. That's only really an issue for games like Animal Crossing, though. And while I have that game, I'm afraid of launching it and seeing all of the weeds.