Am I the only one to think that the IL's are enabling DB? "Jim" throws a temper tantrum because he wants "Jack" to go to a golfing weekend without the wife and kids. Jack wants DW & kids to go along since that's the way it's always been and, besides, "John" and "Dave" are bringing their families.

Jim goes whining to mommy and daddy that mean brother Jack won't do what HE wants Jack to do and mommy and daddy agree that Jack is the bad guy. Mean Jack must kowtow to DB's wishes and, until Jack sees the error of his ways, he is no longer welcome in their home while Jim is around.
Mommy and Daddy - if they were REALLY intent on the boys patching things up - should have
invited BOTH boys to their home and insist that they patch things up right there and then. If Jack or Jim left in a huff with that, then yes, that boy is the child and needs to reevaluate the situation. If Mom and Dad wanted to remain neutral, then both boys should be welcome no matter who is there. If one decides to leave because the other is there or shows up, that's an immature reaction that, as a parent, I personally wouldn't put up with let alone condone.
It doesn't sound to me like OP and dh are the ones keeping Granny and Grampa away. IMHO, the IL's are the ones to start the rift between them and DH and, if they TRULY wanted to make amends, a birthday card ain't it.
They know where DH works - call him. Ask him to meet them to talk. Ask him if they can see the kids. Sending cards is either a way to get under OP & DH's skin or it's a way for the IL's to convince themselves that they're "doing everything they can". Bah!
OP - toss the card. Don't give them any ammunition to say "see - look how horrible she is. SHE's the one keeping the grandkids away from us!". Toss it and don't give it a second thought.