Do you self-host Jitsi? The public instance has absolutely unusable FPS for streaming gameplay which is pretty much the only thing I still use discord for because it’s the only thing that seems to do it well. I read somewhere you can turn up the FPS on a self-hosted Jitsi though.
I've played around a bit with Remotely (https://github.com/immense/Remotely) from what I've seen it's great for when a family member needs some remote support, it also has persistent agents for some stuff. Personally for remote administration I LOVE Apache Guacamole, it's super handy being able to use RDP/SSH/vnc/etc from a browser.
Edit: after reading the post it may not be what you're looking for. I don't think either of those would be good from an FPS standpoint.
What is your issue with Rustdesk? I would understand if you mention Teamviewer, which runs through 3rd party servers, but Rustdesk? Is the client phoning home despite self-hosting or what do you mean? I am seriously interested.
On the other hand, I would not use RDP over the Internet without a VPN or at least an RDP Gateway in between both parties.
There has been a few controversies so here is the TLDR:
Rustdesk public relays have been caught using Chinese servers
The Rustdesk app is highly privileged for little reason. They abused this privilege to switch Wayland users to X silently
very little is known about the Rustdesk "team" if there even is one. There is a single github user called "rustdesk" and the personalities on the Rustdesk website appear to be fake.
Rustdesk says they are from Singapore but there are multiple leads that point to them being Chinese
They straight up delete any issue or discussion that questions the supposed Rustdesk "team"
That's just the summary that I know. You can find more information by looking up Rustdesk controversy or is Rustdesk a scam.
Avoid Rustdesk like the plague for security reasons.
I recommend RDP as a protocol for remote access. It takes a little setup but it can be used. It isn't fit for all use cases of course so don't take it to literally.