There is the avermedia bu113 card, around 100 euros, works on everything from Windows pc, to ipad and of course linux without any additional drivers as it supports with UVC, downside, you basically need a hdmi splitter if you want to see something as its build to use big cameras as webcam
I might have to retract the “flawless” part of my “sales pitch”. I tried to make it spontaneously work on my work laptop using Fedora and it did in fact, not work. To visually show how it should work, I made a picture with an iPad where it actually does work with my work desktop with 2560x1440 resolution, 60fps. But, it did worked in the past, so I am fairly certain I just suck at using my device. If it would work it appears just as any other capture card…as weird webcam device.
But as said, you need a splitter before the card and then one output into the card. I actually spend a minute to locate a non-black HDMI cable just for you! https://imgur.com/a/BvZDZRJ