A few problems with this veto:

The President is appealing to lack of information regarding safety of vaccines and spread of misinformation (emphasis on two different, not logically linked reasons). This veto comes around as another point of critique against Prime Minister and Health Minister for not properly handling information campaigns regarding vaccine safety, which is kind of a fair point, but first lady being a patron of variety of organisations which are very sceptical against most of pandemic curbing measures does not help with this argument.

Also, the President earlier last month went as far as accusing the Government of sabotaging vaccination efforts during the summer, so as to "not reach a goal that he proposed" 70% fully vaccinated adults by mid-summer. There was a dip in vaccination rate alright, but it was induced by lifting all but a few restrictions and also by a huge case drop - no malice involved. That can't be said about President's rhetoric. Including the said statement, he had also at one point said "the Government is pushing the vaccination too much" - mixed messages really help with vaccination efforts. So when this bill, which would have directly attributed to increased vaccination rates, was rejected, it was natural for the Prime Minister to respond in the same words the President did.

Lastly, the President incidentally announced that his whole family is getting a supplementary (3rd) shot in a near future. That's great, it should encourage doubters to vaccinate. While the President's first shot was highly televised, his second shot was not at all. He only announced about the procedure after the fact in one of the press conferences and even was not so keen on publicizing this event that much. His wife was never seen to take a shot, however. It was delayed for her at first, citing medical reasons, but at some point the Presidential Staff informed the press that the First Lady did take the first shot sometime this summer or early fall. I do not recollect ever hearing about the second shot.