Antivax has plenty influence on politics. They've been able to push states to allow more and more capability to dodge vaccines when they should be mandatory unless there is a medically compelling reason not to.

The antivaxx movement recently had an impact on CDC policy, Department of Education involvement at Federal and State levels on how to handle unvaccinated kids, laws passed by State Governors, and two Senate bills. I'm not sure how you're able to minimize their impact on politics.

There are clear links between false statements, the antivaxx movement and the resurgence of deadly medical conditions.