On the contrary. Your time will need to be devoted to your children all of the time sure, but I mean...my son is about to turn 6, and we've had some real bonding moments playing games together. Alternating lives on SMB games, playing co-op on whatever we can, etc.
He even saw me playing King's Field (of all things), fell in love with it, wouldn't shut up about the MUSIC (again, of all things), and begged me to play it through for him. Now we're on KF II. And my wife plays as much as she can, though she doesn't game hard at night after the kids go to bed like I do.
Those multiple sources were blowing smoke up your ass.
I find that many, many people have this mindset that they have to martyr themselves when they become parents, as if they cease to exist for anything but childcare anymore. My wife and I both agreed back in November when we found out that we were pregnant that we would always be sure to put ourselves first and our family a close second. Helicopter parents are weird. How can you parent well if you're totally stressed all the time because you make no time for yourselves?