The picker now leads the form in both clients and takes focus on open, so
Tab walks from it into the command tiles. Opening the form also preselects
the project of the session you were last on — a worktree session seeds its
base repository — so 'n' then Enter spawns another session there.
Spawn similar dropped the session's cwd into the free-text directory hatch,
so the project picker showed nothing selected. For a worktree session that
cwd was the daemon-made checkout, and the worktree toggle defaulted on, so
the clone asked for a worktree of a worktree.
It now seeds the picker: the cloned project's box selected, the same
worktree decision, base repo plus '+ new' when the source had a worktree.
A seeded path that matches no project falls back to the raw input showing
that path instead of an empty picker.