Nah, the rota is large enough that it will likely be somebody else’s problem anyway and the chances are even if it does land on them they just won’t answer the phone.
Punishing mistakes with unpaid overtime has never been a good approach to quality. It just teaches management that they can get away with low quality because the engineers will pick up the pieces in their own time.
Through European lenses this part seems insane. It is work, so pay me for it :) Every oncall rotation I was part of ever was paid, is the "unpaid" part a US thing, or was I just lucky?
In the US it's common to either negotiate 'differential' pay for the responsibility, or as one might see in this thread, get suckered into it for free.