If you have a GCN with the digital out port (Dol-001) you can buy a GCVideo adapter like the Retro bit prism to convert the signal to HDMI. That's the best way to play on standard equipment.
Retro-Bit Prism HDMI Adapter for GameCube - AV to HDMI Converter/Upscaler for 1080P Support https://www.amazon.com/dp/B08WBQL4VM/ref=cm_sw_r_apan_i_WRW9P1TN0H7R95XDCJTC
Thanks, it looks like the GCPlug is like $60 now and the Prisma is only $80 with prime shipping. I’ll probably go with that, thank you!
This or this if your console is a DOL-001 model. That will get you a "lossless" conversion to HDMI, which should help substantially. If you want near emulator-quality captures, you can combine one of the above adapters with an HDMI-to-component converter and a RetroTINK-5X or OSSC. For reference, here's a capture from my setup, which is GBI-HF 360p -> Carby -> generic HDMI-to-VGA converter -> OSSC -> USB3HDCAP.
It does what its supposed too. Had it for ~3 years now. Not exactly the prettiest, but it sits behind the console and not like I see it. www.aliexpress.us/item/3256804373786187.html
Here is one on Amazon you may prefer. I believe the Carby is around the same price: https://www.amazon.com/Retro-Bit-Prism-HDMI-Adapter-GameCube-Converter/dp/B08WBQL4VM/ref=sr_1_3
That's a very long post so I may have missed something in it, but it sounds like you just need an Extron Crosspoint and some RCA-to-BNC adapters. There are S-Video-to-BNC adapters as well, though they're not quite as easy to find.
As for your GameCube, throw the Hyperkin garbage in the trash and get a Carby or Prism HD. If you need analog component output, a generic HDMI-to-component converter will work fine. This is assuming your console is a DOL-001, though. If it's a DOL-101, you're stuck with composite or S-Video/RGB depending on region.
You're better off with a Retro-Bit Prism: https://www.amazon.com/dp/B08WBQL4VM/
