For proof of per-existing conditions....
How would we have gone about that anyways? Would we have had to get a doctor's note, who pays for that, what stops a doctor from just giving everyone who asks a note? Maybe your doctor automatically sends a list to the government of patients at risk, what about privacy, what if the doctor misses someone and they get denied and then die of COVID? Do we just give the "system" access to everyone's health records, how much is that going to cost to process everyone and how long will that delay the vaccines? What about people that do not have a doctor (very common), all the above is broken for them. Personally I do not see how this can be enforced and why would doctors or anyone else want to be at risk of lawsuits etc.
IF we get all the vaccines they claim we will get in the next few months it is also a bit of a moot point because prioritization is only really required when there is a real shortage. They are doing prioritization now based on age and occupation, those are easy. The problem they have is they started down the path of prioritization (needed to) if they just open up the gates then the usual people will be all confused why did we do it before (same people still confused over masks and distancing....).