RustDesk sort of fits the bill. It's open-source, has 2FA, can be self-hosted (but not needed), the client runs on anything, but the main issue here is that no amount of workarounds will make an untrusted machine any less untrusted, you're essentially extending the display and input from a dubious machine into your own.
If you're really worried about the security aspect, my suggestion would be to only use your phone as the client, and if you need to do anything more complex, use a Bluetooth keyboard connected to it. There are some foldable keyboards that don't take too much space and are not terrible.
If you have an interest in Arch, I'd recommend starting with a derivative distro like EndeavourOS. It'll give you an easy installation process and a desktop that's ready to use.
Then just use it as your daily driver. You'll eventually run into the occasional issue when package X or Y upgrades and breaks something, learn to fix that, and eventually learn the "ins and outs" of Arch. That's how I started, I went from Mint to Antergos, used that for a while, then when Antergos was discontinued (RIP) I converted my install to "pure" Arch and never looked back.