Even if they weren't, most MPs will be old enough that their top level Birth Records are available, so if they don't have a common name or you know enough biographical details you can find them anyway. If they've got a weird name like Elon Musk you just go name -> DOB in one step, if they're a Sarah Black maybe you need to know approximate age, birth name, rough part of the country where they were born. For super common names like John Smith you will need to know the mother's name, year and specific location of birth at which point yeah you're closer to just knowing their DoB anyway.

The Online Safety Act is completely stupid - it just requires "age verification". Various sites have outsourced this to various providers, most of which are American. The age verifiers in turn just look at the image, generally algorithmically without human intervention, and go "seems legit". There is no single national UK ID database that could be used for this purpose - everything is either partial, doesn't include photos (e.g. electoral roll), or isn't public (passport DB).

It's all just random third party shit with zero government oversight. There's no national ID in the UK because even the mildest mention of a voluntary national ID scheme leads to gnashing of teeth and wailing about how it's fascist and a violation of our civil liberties and everyone is happy pissing about with driving licenses or passports or utility bills to prove who you are.