You're not going to fix anything by taking a negative approach. Of course you know you need to apologize for the juvenile ultimatum.
If you want your child to see his grandmother more, you and/or your husband just need to take him over there more, coordinating things with the grandmother and the sister.
Let your husband step in and deal with things, especially if he's still working through his own issues of feeling left out as a kid.
Don't burden your child with any of this baggage. He just needs to spend time with and build a bond naturally with his grandmother, aunts, uncles, and cousins.
You may end up having to accept that, for whatever reasons, your child won't be as close as you'd like to those relatives. That's just how it goes sometimes, and you can only do your best to facilitate the relationships.
A lot of it has to do with her relationship with her son. I have the same issue with my MIL, and I had to give up on her.