If you want a no-code solution, I recently created a homepage using GrapesJS (for free). I’m hosting it on Cloudflare Pages (for free). The whole setup was dead simple and almost completely free, I’m only paying for the domain.
EDIT: oops, that isn’t technically self-hosted…but GrapesJS is a very cool tool for building a simple HTML website. Just make it looks like you want and it’ll spit out all the files you need for hosting wherever your heart desires. Caddy, GitHub, whatever.
BeyondTrust and SimpleHelp both support this functionality. You can invite a second technician to join the session. I suspect all they’re really doing is opening up a second session at the same time, but the end result is that you can have two people remotely controlling a single computer.
But those are both “tech support” focused tools, so I don’t know if they quite fit your use case. Parsec is another option that specifically advertises collaborative work like pair programming: https://www.youtube.com/watch?v=XA21BnUsBJI