Those people who do not trust the government are not the target audience for such actions. They are trying to lure those who are hesitant or just lazy to get vaccinated. Based on the example of my country, I can say that the effect of such actions is about zero, and maybe even negative - all people understand that free cheese is only in a mousetrap.
A little carrot and stick method won't hurt, I mean if it does the job, we have no questions about it right and I think that it's effective since it's making people encouraged to get vaccinated and given how a lot of people loves lottery, who would refuse a free ticket just for some doses.
The problem with vaccination revolves around the people and the media spreading misinformed news about it. Unfortunately, these people already have their biases towards vaccination and this fuels the fire more when they see people spreading fake information about it.
The government is trying their best to limit the movement of unvaccinated people by giving incentives to the vaccinated. In a way, this kinda forces these unvaccinated people to get vaccinated due to the rewards that they will be receiving once they got shot.