The £10 cam link: I bought a cheap USB HDMI capture card, and it works!
I have EposVox (https://www.youtube.com/eposvox) to thank for the heads-up, after he made a video about a mysterious unbranded USB HDMI capture card that seemingly appeared overnight, and defied all his expectations. He's nicknamed it the 'Can't Link', even though it definitely can. Mine didn't come with a USB extension cable, though.
My follow-up video on audio capture from the £10 cam link: https://www.youtube.com/watch?v=Q5K8UBqd14o
EposVox's video: https://www.youtube.com/watch?v=daS5RHVAl2U
eBay suggested search (not affiliated): https://www.ebay.co.uk/sch/i.html?_nkw=hdmi+usb+capture+card+1080p&_odkw=hdmi+hd+usb+capture+card
0:00 Introduction
1:46 Unboxing
3:08 Hardware setup
3:25 Software setup and first camera capture testing (OBS Studio)
6:22 More camera capture testing: two at the same time?
8:16 Capturing computer output (from a Windows tablet)
9:20 Conclusion
I'd actually completely given up on getting a HDMI capture card, as the recommended ones (even second-hand) were ridiculously expensive. I don't need 4K at 60 frames per second: 1080p at 30 frames per second is more than enough for what I need right now, and spending over £100 to get that isn't really justifiable. £10, on the other hand...well, I bought two, and nearly bought more!
This is a different design to the previous cheap (around £15) USB HDMI capture cards that I'd seen, and which had less than great reviews, with users reporting that they didn't capture in full HD and/or they overheated. These new ones seem to have a lot higher build quality, with a screwed-together metallic case.
I've only just noticed that there does seem to be a slight colour difference at the end, where I recapture the video I recorded about an hour before, but I can't tell if that's the capture card itself, or the way my Windows tablet (or Windows Media Player, actually) has decided to render the colours. To my eyes, the colours look perfectly natural, and if there is a slight tint, I'm sure that could be colour corrected in OBS.
Please note: I have not been asked to endorse any of the products in the video, nor am I receiving any sponsorship from the companies who have made these products. The video is purely based on my personal experience, the outcomes of my own testing, and my opinion as a result.