Does the eligibility information even go to the insurance company?
Yes- you have to either submit the form online and bring the submission number to the pharmacy or fill out a paper copy at the vaccination site.
Although as far as I know , pharmacies in NY are only vaccinating those who are eligible due to age and I believe that is a group that must provide proof of eligibility. Everyone has to fill out the form, but you might also have to provide proof of eligibility depending on the reason you are eligible.