Very recently UK medics noticed some highly deleterious effects of Tetra Fluenz flu vaccine and pushed back the earliest age for its use to 24 months. They not only entertained not vaccinating, they advise it.

It's irrational to not consider the question of whether taking a vaccine is right (for yourself [or dependents], and for the population).

There's a big difference between a flu vaccination and MMR.

Not as big as you are thinking the MMR is normally given after one year. This is because there are some know potential negative side effects when babies get it earlier.

The point is that society already evaluates these hazards/risks very effectively so that individual parents don't need to. Indeed any individual parent's risk analysis about vaccines is likely to put their children at greater risk.