I've worked at places with this, usually called something like rotating team lead, or duty engineer, or ops duty, or pager week, or something like that. To the best of my knowledge everyone involved dreaded when their week came up. I've only experienced this implemented WRT (metaphorical) fire fighting not traditional dev work. If an earthshattering bug appears this week, you'll be fixing it and/or triaging it and/or figuring out who can help you fix it.
Its awesome at getting rid of "pass the buck" and awesome for everyone else in the org to have a single point of contact (even if its a different human each week). Not so awesome around changeover time or if your docs aren't up to spec or when you discover the hard way someones code isn't good. Also not so hot at fixing major architectural issues (like, its going to take more than a week to fix this correctly, and I'm only on duty for a week, so whip out the band aids...)
Rotating through on call, especially having not just ops, but engineering rotate, is a huge benefit.
As much as DevOps is being promoted, there are still two fundamentally different skillsets and job descriptions, and if its just the Ops side of the house getting the 2am wake-up pages, you're going to get a decidedly different attitude toward Things Wot Breaks Prod than if the developers are on that rotation as well.
Plus it means rotation comes up that much less often, which is a good thing.
Its awesome at getting rid of "pass the buck" and awesome for everyone else in the org to have a single point of contact (even if its a different human each week). Not so awesome around changeover time or if your docs aren't up to spec or when you discover the hard way someones code isn't good. Also not so hot at fixing major architectural issues (like, its going to take more than a week to fix this correctly, and I'm only on duty for a week, so whip out the band aids...)