Isn’t this basically the same problem as the need to update the DISPLAY variable when X11 forwarding is enabled?

Why not a more general approach like this?

https://www.johntobin.ie/blog/updating_environment_variables...

This is explained in the original article linked in the first sentence: https://blogsystem5.substack.com/p/ssh-agent-forwarding-and-...

The image? Yes. The text? Not at all.