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.
One factor is those negative news or misinformed news that keep spreading regarding to the available vaccines, the government is indeed trying their best to get everyone and have much lesser chance of spreading the virus, with this lottery some unvaccinated might try their luck and participated in hope that they will win and have a chance not just to move freely but also to win huge amount of money.
We can't go back to our normal ways of living without being vaccinated, just my own cents. This virus will stay and we need to do our part to prevent suffering from this covid virus.