Regarding the importance of your CDC card or it's validity....
Both my wife and I have had our vaccinations and have our "cards". My wife's is one they printed on copy paper, she had the first shot at one Walgreens, the second at another. Her name, address (my daughter's since she was there during that time) and birthdate are handwritten in, as well as the Vaccine lot, date and location (sort of, kind of legible). Mine has the advantage of being an actual card, but I had to fill in most of my info. My wife also works in a pharmacy and administers and processes info on vaccines everyday.
To begin to think that the card is any proof of vaccine is laughable. The form can be printed out on any home printer, and the information is hardly secure in that you can make up about any of it with no problem. As reported in multiple news sources, there are a number of places selling fake cards...you don't even need to do that if you have a printer and a pen, and maybe an understanding of valid Lot Number format.
The gap right now, is the link from you to verified health records. That is where HIPA records apply. A card is nothing, but someone who can look at you, then have valid access to health records, just does not easily exist. Yes, you could be asked for your CDC card, and if they want to accept that, fine...but figure that if the Yankees all of a sudden say...hey you have to have a card to come to the game...who is that going to stop? They can not verify your card, just either accept or reject.