Depends on what you want for your future I guess. If I was in your shoes and I knew I want something different for me in the future, I'd break up (maybe stay friends with benefits) and look for that future. Otherwise I'd be scared to miss out on that person.
Your are a grown up and obviously know that this won't work out long term so stick with it as long as it makes you feel good and doesn't prevent you from working towards your desired future.
Having friends of a different gender is totally fine. I'm a 36 year old woman and I'm with my spouse for about 7 years now. My best friend is a guy I know from uni. His gender wasn't even a topic to begin with.
But
Makes me icky. Does your boyfriend really think this is the only reason a guy (or anyone) could want your friendship?