He didn't blow it though, that scene was more than a little poorly written.
If you literally don't think you have any family left then there is nothing wrong with believing that belonging to Star Fleet is all the family you need, plus who knows if he really even meant it? In the context of that scene he's more or less in graduation speech mode amping up the new recruits, so of course he's going to be "ra-ra! Go,Federation!" and Jack had been watching the entire thing so he would have known that. If we were meant to see that as Picard screwing his relationship with Jack it didn't work and I doubt it was there to make Jack look petulant so the only answer is poor writing. It happens, even when a show is otherwise pretty fun like this season.