I've been using a Sony mirrorless (anything above a5100 will work) for over 6 years now; it needed a "dummy battery", and an HDMI capture card (about $25 for noname brands, or $80+ for Elgato, BlackMagic etc). It auto-focuses, doesn't write to microsd, and works flawlessly.
Even if you aren't buying Elgato, you can use Elgato's compatibility page to know which cameras work well: https://www.elgato.com/us/en/s/cam-link-camera-check
Weirdly I had the exact opposite experience. Elgato always felt laggy. I bought a no-name USB Stick format card and it looked great (once I got my camera settings dialed in) but would disconnect when I bumped my desk. I cracked the case open and soldered a USB cable I cut in half to the pads, and 3d printed a new case and it's been rock solid for the last 4 years. Only problem is the once in a blue moon I need to use Teams my video get's horizontally squished and I can't seem to fix it.
A word of warning on capture cards: I first bought a no-name off Amazon, thinking to save money. The video quality was abysmal. Artifacts everywhere.
I returned it and got an Elgato, which has worked great from day one.