hm, that really nice solutions - what capture card do you have?

I’ve done this the opposite way round, using a cheap USB C capture dongle from Amazon to capture output from an RPi and then using my a Mac as a virtual screen for the RPi. I do this occasionally when I need to setup something but don’t want to drag out an external monitor.

The capture card/dongle are designed for using in Twitch game streaming where you are using it to capture the output from a game and feed it into something like OBS. The one I have was pretty cheap, but only supports an output resolution of 1080p (input up to 4K). I don’t remember the exact brand, but the above should give you enough keywords to get started.

Elgato HD60X: https://www.elgato.com/en/game-capture-hd60-x

There are cheaper options if input lag isn’t an issue.