Op your don probably has a higher need for play dates than this kid, so you really are getting a be for you aren’t acknowledging. The other boy might be neutral or equally happy with one play date per semester but also fine to go if invited. My guess is your son bugs you if you don’t have a bust playdate schedule and that’s your burden, not the other families.
My son likes seeing his friend, but honestly, is ok just spending time with his own brother. But of course, his brother can't be his only friend. Often, the friend's mom will ask me if the kids can get together - but then always asks if they can come to my place or asks me where I think they should go if I can drive them somewhere (public transportation is not good where we live).
I think the issue is two things:
1. Her being pushy and putting it on me to either come up with an idea, host, or drive the kids - which can be handled me by not agreeing to do this all of the time, only sometimes.
2. My son objectively wanting more friends (so he says), but not really being willing to put in the effort, and me feeling sad about that - which has nothing to do with this family, but is probably increasing my negative feelings.