No they don't. You use the same hub to hub model as airlines. E.g London to Edinburgh, not London to Edinburgh stopping in 4 places.

One of the nice things about traveling at night is you have less time pressure and congestion so when doing track repairs it's usually fine to divert the train.

Hub to hub - does that mean you wake everyone at 3am to change trains? Nobody switches cars with humans in them.

Yup. These two points are not an issue at all, and are in fact strengths for sleeper trains. Being able to redirect a sleeper train without any (or minimal) impact on its timetable is a big plus.

Usually they depart from the last boarding station around midnight, and the first disembarking station won't be hit until six in the morning. Some outliers exist, but the number of people getting on or off there is negligible. For most travellers you are in the train before 22:00, and won't leave before 6:00.

This ÖBB line is typical: https://www.nightjet.com/de/reiseziele/oesterreich/wien

Yes, you can get off in Nürnberg at 4:08. But almost no one does that. The train just happens to have a halt there¹, but 95% of the passengers get on in the Netherlands and the Ruhr Area, and won't get off until Austria (and vice versa).

1: I suspect mostly for rail topological reasons.