Sure would be great if we could get to Denmark's vaccination numbers.
This is like saying "We should stop all the DUI checkpoints like this other country did," when that country doesn't have a DUI problem but 30% of drivers in our country are drunk.
You're talking about the UK, right? From what I understand they're still going to ID you one way or another (e.g. they'll just look you up in their database, just like police in the US will do if you don't have your license on your person).
Sorta works that way in Canada too. At a DUI checkpoint where they pull over everyone without reasonable cause, the stop is supposed to be strictly to check for DUI. They’re not supposed to go poking around for more things unless it’s incidental to the stop (if they see a dead body, they can investigate that, but not supposed to decide to check your immigration/license/insurance status as a part of the stop).
They'll ID you if you did something wrong. If not, for instance if they're randomly pulling over cars to check for drink driving and you're clean, then they don't need to ID you.