Better than that would be for our society to get over the stigma around STDs and STD testing. As unromantic as it is, it should be uncontroversial to ask someone for a copy of a recent STD test. Even having cards would be nice, on a purely opt-in basis.
People will whine and moan about the privacy concerns, but when it comes right down to it people with untreated or untreatable STIs should not be having sex (and if they are their partners should be made aware.)
Actually, it's worse than that. Herpes and HPV can be transmitted via skin not covered by the condom, so even if the condom doesn't break it doesn't necessarily prevent transmission. It only reduces the probability of transmission.
FWIW: I decided to become celibate after the third moron in a row failed to comprehend the statement that "A condom is a necessity even if pregnancy is not a concern". I don't have an STD. I have a compromised immune system and was extremely ill, with one or more antibiotic resistant infections. Doctors never identified the infection(s) in question. I concluded that no amount of explaining was going to convey to anyone my belief that they were at risk of exposure to infection (doctors chalk it all up to my genes, so other people go with that mental model and assume that although I can get sick from them, they are somehow magically at zero risk from me) and it would be best to just not go there until I was healthier. (I also concluded they were acting with callous disregard for my welfare, a very big red flag in my book, but that's not really relevant to my point.)
So while I agree with some of your points, I really don't think a card or test will address the issue. For one thing, if someone asked me to produce my card (or latest test results), to me that would just scream "This is just a hook-up. You are nothing but a sex object to me." At which point, if I did the traditional dating thing, I imagine I would get up and leave the table. For another, different people have different levels of understanding of what the risks are. So I would question what it means for their partners to be "aware of the risk". Do you just give them notice of your state of health or do you have a larger obligation than that to ensure they REALLY get it? Just giving men notice of my state of health struck me as wholly insufficient. They blithely climbed into bed with me anyway (and without a condom), apparently oblivious to the potential implications of sleeping with someone who was deathly ill with something doctors did not know how to effectively treat (or even really identify).
Actually I would wine because I now have to go to the doctor just cause some idiots can't remember to use a condom (condoms doesn't break very often, or else prostitutes wouldn't insists on using them).
People will whine and moan about the privacy concerns, but when it comes right down to it people with untreated or untreatable STIs should not be having sex (and if they are their partners should be made aware.)