It's a bit more complicated than this (disclaimer: I followed the trials and vaccinated earlier than most).

The current wave of anti SARS-CoV-2 vaccines skepticism has multiple facets and is also country dependent.

First, the risk profile of the disease meant that the best solution would be vaccinating those at risk immediately and leave the rest for judgement of the individual. However, people took the infamous Israeli study about 94% reduction of risk of asymptomatic infection (hence transmission affecting) at face value (the 5MM people from that study weren't swabbed regularly; it was an extrapolation), which meant that some tried to follow the dream of impossible elimination (impossible since March 2020) so they thought everyone needed to be vaccinated. Previous evidence on other CoVs was largely ignored.

Second, many vaccinated to go back to a normal life. Imagine the reaction when those people were faced with more lockdowns, masks, restrictions... Some thought they were made fun of.

Third, the government officials absolutely hated good news as they thought fear was good to achieve compliance, at least in UK and Italy. In Italy, not the nutjobs, but respected members of medical society unwillingly questioned the efficacy of vaccines ("half-work", someone said). This confused the general public even more

Fourth, adverse reactions are to be expected and with loads of inoculations, some are bound to appear even if very rare. Even more with these vaccines which are highly reactogenic. The confused reactions on the AZ vaccines when there were cases of thrombosis were eloquent. Also, risk vs benefit ratio was not considered. Children have very low risks but have a higher incidence of (rare) adverse events. Given this, the ratio isn't favorable, but it was pushed anyway without solid data (the confidence intervals on FDA documents for pediatric versions of these vaccines are horrid).

Fifth, some countries went out of their way to make the lives of those not vaccinating very miserable. In Italy you could not even work without three doses. Children without three doses > 12yo could not take the bus legally. And the whole Canadian truckers matter.

While antivaxxers are a very loud but small minority with no real power (they're extremists) the whole matter was a massive failure at scientific communication. Also because governments wanted to treat citizens like stupid peasants that needed guidance by the enlightened. Well, not all of them, but in Europe there were many.

Yes I'd agree that it is impossible to prove something to someone you don't share common ground with. I'll never be able to prove ghosts don't exist to someone who believes they do. I might however be able to prove the absence of any evidence for the existence of ghosts, provided that the person I'm speaking to has the same perception of what constitutes evidence.