For remote access I use two solutions that both have the features your looking for.

The first one which I have used for longer than I can remember is MeshCentral github.com/Ylianst/MeshCentral

This one works great, is web based, let’s you access a shell, or you can make port tunnels been agents, access RDP or vnc in the web UI, the file system etc.

it’s a really great one stop shop that more than covers what you’re looking for.

The second is RustDesk github.com/rustdesk/rustdesk with the betterdesk server and web UI github.com/UNITRONIX/BetterDesk

This one scratched my TeamViewer itch and weekend pretty well. It’s newer than MeshCentral and not as easy to setup however on top of the web client it’s got apps which in my opinion provide a better experience for remote desktop access than MeshCentrals novnc client especially around copy paste.

All in all, of your looking for something ultra easy I recommend MeshCentral, it’s minor quirks around novnc are so easy to overlook since it just works so well.

If you want a modern TeamViewer then go with RustDesk/betterdesk but it’s newer and needs some extra work to get going.

All these projects are open source, free, self hostable, has headless file and terminal access, and have web clients.

For using Linux on any remote solution you pretty much have to run the desktop on x11 instead of Wayland. I don’t know if any software has found a great Wayland solution yet.