This is a simple organizational problem with a simple solution: don’t do that. Instead, ensure everyone on the team has a production pager, and that the person responsible for work prioritization gets every single page in order that they feel the pain of that (this may well be the lead, like you say!).
Align incentives to create simple, reliable software.
> and that the person responsible for work prioritization gets every single page in order that they feel the pain of that.
This sounds like an incredibly toxic work environment
Having one single person get all the pages does sound like it's going a bit far. The point though is for there to be a tight feedback loop between problems in the infrastructure and the ability to solve them. If this isn't the case, someone else in the organization is feeling that pain, but often they aren't empowered to solve them.
It’s substantially less toxic than the people who prioritize work only prioritizing feature delivery, and not operability.
I'd recommend the lead takes the role.