We already have that.

On a spectrum of options, no verification is the least privacy intrusive. Baking it in at the OS level or forcing passport uploads are the most intrusive. My proposal is in the middle.

A determined actor could maybe follow you to the store when you purchase your verification code, take a quick picture with a powerful camera (or bribe the store to do it sneakily) and unmask you online. But there's no way to do it at scale. And if you buy the code from a reseller (ask a panhandler to buy one for you, perhaps) then it's even more robust.