Usually the argument I have heard against them is the second order effects.
In any individual incident, a helmet would make things better. However, a helmet is a very bulky thing, most bikes lack a secure storage, and people just don’t want to carry around a big helmet when they go shopping, or to a restaurant, or any other activities. So if you mandate it, biking rates will go down, and cyclists become so uncommon on roads that drivers stop being familiar with how to drive around them, stop looking out for cyclists that are almost never there, etc.
There’a also some evidence that drivers will adjust their behavior based on perceived cyclist protection and skill level; the behavior around a person in a hat and sundress is different than someone in full Lycra with helmet.
But jurisdictions with helmet mandate show no clear reduction in injury rates: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1410838/?tool=p...