I'm a stills shooter who purchased an R5 C to explore video. What I do capture, typically 4k 24p XF-AVC goes to a card. You can however capture video from the HDMI out on your EOS R. (4K C-Log in 10bit). I have not used any Elegato products but have followed their tech over the years.
I found a bunch of examples where people were recording directly to a ninja V. If you press the record button on the camera, a card in the camera is required, if you start/stop recording from the ninja, no card is needed as it records straight from the HDMI out.
OBS Studio will capture video. I just read the overview. You can set a hotkey on the PC or MAC to start or stop recording. So, if you connect your R to the Elegato and use OBS you'll be able to capture video. You may want to review the Elegato specifications. I think it allows 4k 30p and 1080 60p. I'm not the video guy here. Either @RS-EOS or @AtticusLake can likely confirm this for you.
Bay Area - CA
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09/26/2023: New firmware updates are available.
08/18/2023: Canon EOS R5 C training series is released.
07/31/2023: New firmware updates are available.
05/31/2023: New firmware updates are available.
05/18/2023: New firmware updates are available.
03/30/2023: New firmware updates are available.