And the odds of a vaccine being literally "mandatory" in democratic countries are essentially nil. No one is breaking down doors and holding families down to get their vaccines.

(What is possible is that schools and even workplaces require proof of vaccination once one is widely available if there is still a widespread virus risk. That could arguably be fairly coercive but it's important to distinguish from being "mandatory.")

Why is it important to distinguish things required by most schools and workplaces from literally mandatory ones?

It isn't since working is practically mandatory in our society and going to school is even legally mandatory in many jurisdictions.

While a widespread vaccine requirement to be physically present in places that will be important to many is a fairly big stick, it's still a far cry from vaccinating people by force. Certainly I might well favor the former but would be horrified by the latter.

That is a very subtle and impractical distinction, similar to making an argument that you are merely being pedantic instead of deceptive :/.

I also think many people in high risk categories (age, preexisting health issues, essential or health care workers, teachers, other professions involving crowds and risk) will be very eager to vaccinate. It doesn't have to be mandatory for high demand and high use.