That's what I've been doing with termux, mosh, and tmux.
Yeah the remote control featureset is pretty limited right now. I did a comparison here https://yepanywhere.com/claude-code-remote-control/ (with my own project). I'm sure they'll iterate on it. Overall it's such an obvious feature for them to add I'm surprised it took them so long to ship. There are probably at least 50 such projects that people have made (https://github.com/kzahel/yepanywhere/blob/main/docs/competi...)
The one feature drawback of tailscale/tmux/termius is no file upload. And ergonomics, ability to view files/diffs easily, though that's subjective.
Which is so much better because you can do other terminal stuff and you can avoid vendor lock in.
I think a significant distinction between your approach and Claude’s approach is that your approach requires allowing your machine to accept inbound connections but Claude’s approach does not. Claude probably went with the latter to avoid a whole class of security issues and mitigate risk of users having their machines compromised. I’m not familiar with what the new vectors of attack are with Claude’s approach though.