I'm not a trademark lawyer so take this as you will, but the term Medicare has been trademarked since 1957, assigned to the Executive Director of the US Army.

https://trademarks.justia.com/890/00/medicare-89000008.html

Just because you haven't run into a trademark issue yet doesn't mean that you will not in the future. And I don't think I'd want to be fighting the US Army lawyers for the naming rights.

I love your initiative, but you may want to look into the naming issue a bit further (since this is what trademarks are explicitly designed for, to prevent confusion about one product/service doing the same thing under the same name).