My personal tinfoil hat theory (not that tinfoil hat tbh) for why refuse to admit they fucked up and put a photo of a completely different guy out is that they have nothing.
Which, granted, can't even blame them. Your suspect is a white guy that's probably on the younger side that checks notes owns a hoodie and a backpack, a description that probably fits hundreds of thousands in NYC. I'm assuming a guy that went through the trouble of putting words on the casings probably also went through the trouble of removing any potential fingerprints/dna. The fuck do you do with that? Add in the public's seeming unwillingness to snitch even if they did know something...
The opsec of the assassin is pretty solid. I wouldn't be surprised he swapped out his clothing at some camera blind spot to blend in with the city crowds and misdirect the investigators instead of going to Central Park as it was reported.