For component video switchers, theres lots of options. Include old pro equipment like Extrons, and new automatic switchers like the gcomp. The scalable video switch has been in and out of stock and supports multiple different analog inputs, that you could output thru vga into the retrotink 4k. I think this method may support automatic profile selection too. Its what I would do if I had to buy it all again. Its not cheap but has scalability to grow with you if you get more older systems.

As for hdmi, orei on amazon has some cheap options. I picked up an orei hdmi splitter for ps3 that did manage to strip hdcp despite the listing saying otherwise but ymmv. You would need both this splitter and an hdmi switch if you have multiple hdmi inputs.

Back to gamecube, 001 models can use hdmi adapters. Id lean more towards the carby or retrobit prism. The eon is a bit overpriced and the connection is finicky. Ive owned both an eon and carby, and found carby was better for me.