I did some research on this since I do not use elgato products but I found these two
"On top of that, I own an Elgato HD60S for streaming on Twitch. Elgato’s capture cards are not yet compatible with M1 Macs, even though they are more than capable, as they require at least a quad-core CPU (though it specifically mentions at least an Intel i5) and at least 4GB of RAM. This Mac mini has an 8-core CPU and 8GB of RAM. However, it doesn’t work with my capture card."
For just $700, this is the best desktop computer you can buy for its price.
bradleyalanlaplante.medium.com
"When I hook up the capture card to my MacBook, it captures the content for up to 5 minutes then it freezes and doesn’t pick up any more frames. (HD60s+)
https://www.reddit.com/r/elgato/comments/kn9554
Surprisingly there is not much information on this but every source that I could find that involved actual testing showed the elgato cards all failing on M1 Macs.
The other topics mentioning this say that in theory it should work through Rosetta 2, however I would put little weight in this due to the above threads.
I am sure that Elgato will eventually update for native M1 support, but for now it might be best to either keep using your current Mac or build a super cheap APU PC for streaming your consoles.