This is definitely one of those "old school" vs. "new school" arguments. It pops up every now and then when contemporary athletes are weighing whether to leave to team or not. For athletes of earlier generations, there was nothing to think about - they were expected to play, and get back home at the next opportunity. It wasn't uncommon for them to miss the birth of pretty much all of their kids. Many expressed regret after their playing days were over though. The contemporary player, with all of the discussion of men getting paternity time, and our more "emotionally in-touch" society, the sentiment is different. Men are now expected to be there to support their spouse, and hold their new-born child. Team management and other players are now more understanding of a player missing regular-season games, but most players probably wouldn't think of missing a post-season game for the birth of a child. Most spouses also understand. They have a pretty good life due to what their husbands do for a living, and learn to suck it up and not complain about something like this. If it's possible for him to be there for the birth and still make it back to wherever the game is in reasonable time to play, teams will usually do whatever they can to help facilitate because it's a great PR move.
"The best kind of pride is that which compels a man to do his best when no one is watching."