ssh/tailssh: break a method into half in prep for testing

And add a private context type in the process.

Updates #3802

Change-Id: I257187f4cfb0f2248d95b81c1dfe0911ef203b60
Signed-off-by: Brad Fitzpatrick <bradfitz@tailscale.com>
This commit is contained in:
Internal Server Error - Gitea: Git with a cup of tea
500 Internal Server Error

Gitea Version: 1.28.0+dev-477-g8b6ad49a5f