What alternative would you propose? Page the oncall whenever there's a single query timeout?

Well, yes. If the cable falls out of the server (or there's a power outage, or a major DDoS attack, or whatever), your users are going to experience that before you are aware of it. Especially if it's in the middle of the night and you don't have an active night shift.

Expecting arbitrary services to be able to deal with absolutely any kind of failure in such a way that users never notice is deeply unrealistic.