I don't see how it can prove citizenship. Unless the non citizen's card clearly says it's been issued to a non citizen. I honestly don't know. Plus, if my card says (which it does) Not To Be Used for Identification Purposes, then it's totally invalid for that purpose. I do have a passport, which ought to be the very best piece of documentation there is to prove my citizenship, although too often that is rejected by idiot clerks.
I'm in New Mexico, and it's been a nightmare here for people renewing licenses ever since the bullshit "Real ID". I renewed mine a month early a couple of years ago in the hopes that when I next renew, they'll have worked out all the kinks. Luckily for me I still get a few bills mailed to my home. People who have gone all electronic, or kids who are still at home/in college/otherwise not on their own have a terrible time. Plus, around here the DMV people regularly give people wrong information or require documentation that is not required. I'd think a Social Security card (Not To Be Used for Identification Purposes) would clearly fit that category.
Although I lived in Kansas myself for a number of years, I was not aware that it issued the birth ID card you've described. I'm real familiar with the Pennsylvania problem.
Anyway, a Medicare card, laminated or otherwise, should never be something you need to prove citizenship, so I'm sticking with my decision to laminate.