Historically liquor store checks were purely visual. These days they are often digital, meaning claims about privacy might (or might not) be outdated. The general principle still applies though. The physical infrastructure already exists, the ID checks do not necessarily need to be digitized or recorded, and even if they are the issued tokens don't need to be tied to the check.

Grocery stores already sell age restricted items as well as gift cards that require activation. The state could issue "age check cards" that you could purchase for some nominal fee. That would require approximately zero additional infrastructure in most of the industrialized world. The efficacy would presumably be equivalent to that for alcohol and tobacco.