Yes, being boosted helps, but a vaccine mandate should be in service to very clear and significant public health goals. Herd immunity is such a goal, but that was likely impossible once Delta hit, let alone Omicron. At this point what is the goal? At best we can only delay infection, but not for very long.

https://www.medrxiv.org/content/10.1101/2021.12.30.21268565v...

This study shows that being boosted gives you 37% protection from infection over a period of maybe a month. But this means at peak spread, you can expect an infection rate among the boosted population of 63% of the unvaxxed infection rate.

If we assume we can get every vaxxed person boosted, you're still more likely to get infected by a fully boosted individual since the vaccination rates are something like 70%. That is, for every 70 vaxxed people there are 30 unvaxxed, and 70 * .63 > 30. If we assume we can get everyone vax+boosted instantly, the estimated infection rate drops to .85 percent of the current rate. With an estimated r0 of 10 for Omicron, such a small reduction in infections has no chance of substantially slowing the spread. At this point mandates are merely a punitive measure against those the majority have deemed moral degenerates.

Slow the rate of infection means I'm infected tomorrow instead of today. There is no stopping it.