Do you sign for the aircraft when you go flying Step? I usually don't.
Tech log is a fair point - but retrospective, and actually as an instructor it's not unusual to have the student fill out the tech log for practice. Signature, yes, IF that's required.
But, to be frank, any flight where it's not absolutely clear who the captain is, before the aeroplane moves under its own power, is doomed at the very least to suffer confusion, and very much at risk of accident, before it starts.
G