“Take the morning if you need it” puts the judgement on the person who was woken up, at the moment they are least able to make it.
the policy, written down before the rotation started:
paged between 23:00 and 07:00 → the next morning is
off. automatic. no request, no approval, no
assessment of whether it was bad enough.
it is entered in the calendar by whoever runs the
rota, not by the person who was paged.
Making it automatic removes the social cost of claiming it, which is the entire reason discretionary versions do not work — nobody wants to be the person who says the incident was bad enough. Someone other than the affected person entering it is the detail that makes it stick. The cost is real and is much smaller than the cost of a rotation people leave the company to escape.