Why don't they just have a Covid vaccine mandate?

I’d say…

Fear of civil unrest.

Western European countries value personal freedom and are loathe to mandate vaccinations, they hope people will come around to it themselves. If they lockdown for unvaccinated (I had no idea that was happening) then the unvaccinated have a choice to stay home are get vaccinated.

Governments are trying to manage the population. Managing means dealing with personalities and trying to do what will be the most effective with the least amount of havoc.

Different countries have different governing restrictions. In the west there are a number of countries with rights similar to America’s bill of rights. But there are others that have more governmental control over citizens’ lives. So, you can’t ask such a question with no regard to what the government is allowed to wherever you are. The world is full differently governed peoples living in highly differing cultures. As it should be.

It depends on where you are and how the gov’t is structured.

So by “they” you mean every government on the planet, for everyone, in all situations?

This is a ridiculously broad and poorly-framed question, and you’ve asked so many irrationally-founded questions about related topics in the past, that it seems pointless to attempt an answer here.