This is what we do in 24/7 network operations centers - so issues can be cleanly handed off between shifts.
However I don't think the additional staffing costs will fly, you're talking a hugely substantial increase if you were to enforce this - more than 25%, since current staffing levels are so thin.
(not sure who would fix this) but that still sounds like a problem with the cost structure. In other businesses, after 8 hours you get paid 1.5x, after 12 hours 2x. If those were the incentives here, it'd be cheaper to overlap and pay two salaries (I'm ignoring benefits here).
It's a balance of what people are willing to pay, the risk we're willing to accept, how ethically we treat workers (doctors). Unfortunately, like many places in the economy, there aren't rational actors on each side to negotiate these things.
However I don't think the additional staffing costs will fly, you're talking a hugely substantial increase if you were to enforce this - more than 25%, since current staffing levels are so thin.