We run a multi-hundred person team here for a live, 24/7 product, and as many as half of our developers have been scheduled as "on-call programmers," which we call our Live team. Their sole responsibility is the live, deployed product and customer-impacting issues.
They do no bug fixes outside of that. They do no feature development outside of that. There is an entire other team dedicated to those things, and like 37s, that team gets rotated through.
We also have QA dedicated to the live product, Operations dedicated to the live product, etc., etc., all separate from new feature development, because an immediate, customer-facing issue requires different prioritization than feature development.