A .22, no, not likely.
The 9mm would do some damage if the bullet travelled through the facial bones at the front of the head, but would be slower to kill. A round to the roof of the mouth might do that too if it fragmented.
Short of jacketed hollow points or a frangible round, the damage would likely be mostly distortion, not destruction if the bullet or fragments did not exit the face--an 'exit' wound.
Something on the order of a shotgun is needed to 'blow the face off'.
I’m not sure where the story came that he had his face “blown off”. I can’t find that in the official press conference information. So maybe it was an exaggeration.
People were looking for a reason they didn’t identify the suspect, and it could be they didn’t WANT to leak his identity because they wanted to question other people he might have worked with.
They may have only released his identity today because the gun shop owner leaked it last night to the internet and it was gaining wide circulation.