The burden of baby sitting.... I would have thought that the appeal of gaming once you are a parent is the valuable adult time AWAY from said children.
What is wrong with getting a babysitter if the kids aren't old enough to play or entertain themselves?
Babysitters are nice, if you can (A) afford them, (B) have someone you can trust, and (C) are absolutely sure how long your session is going to run.
Point A is a sticker for a lot of new parents. Sitters have come a long way from when I was a kid and they got paid a few dollars an hour. Now they want upwards of $5 an hour per child, plus money for food.
Point B is a judgement call. We are a lot picker than some parents, and have real issues with just handing over our kids to random person we've hardly met. Add into that having someone rummaging through your belongings, running up your phone bill, ordering movies without you knowing, or whatever other nightmare scenarios you can think up, and it can be a sticking point.
Point C ties into A. I'd be nice if a session ended when you thought it would, but things get going and everyone decides to go another hour, or you lose track of time, or whatever.
If both my wife and I are gaming, we have no problem hosting the game at our home. That frees up the babysitter money for ordering pizza or getting drinks, and the minor inconveinence of the kids wandering in to see what we are doing or asking for something to eat or a new TV show is well worth the peace of mind.