So I think we’re thinking along different lines – I’m thinking that this is something that browsers could just do – and I think you’re thinking that this is something sites would request.
Browsers already differ a good bit – for example Chrome remembers WebRTC microphone/camera permissions (but has constraints on when sites are allowed to prompt for them, requiring user interaction first) whereas Firefox asks for microphone/camera permission every time on the theory that just because a user did a videoconference once doesn’t mean they’re comfortable showing their current state to others (but is more lenient about allowing them to prompt).
It’s not clear to me what would give sites an incentive to ask for a time limited permission.
But I can see why a browser maker (or browser extension author) might think it’s the right thing for their users to interpret a user’s grant of permission as “for a day” rather than “just for this session” or “for all time”. (And browsers already do both of “for this session” and “for all time”, and sometimes already differ between browsers.) So I think this is something that browsers can just experiment with today – it’s not clear to me what APIs it’s not compatible with. But there might be some, and fixing them would be reasonable.