Vaccine mandates and the threat of vaccine mandates in the US in particular have successfully driven up vaccination rates and allowed removal of pandemic restrictions earlier than would have been possible otherwise.
The studies show that giving better information helps to undecided. Whereas vaccine mandates do not help much for staunch antivax.
The real world example (for example, the UK) also shows this in practice. No vaccine mandates are really needed to ensure high vaccination rates.
Vaccine mandates may have improved vaccination rates in certain groups but those groups are not most at risk. The total impact on vaccination rates and public health may even be negative as with more elderly have refused vaccine due to mandates than without the mandates.
I think you're discounting an incredibly powerful distortion field created by malevolent misinformation by some very sick and confused people.
Vaccine mandates however can caused considerable friction in the society. Just see Canadian convoy protests with hundreds of millions of damages.