My dad works for the airlines, and he said that, outside of major delays and cancellations due to weather, etc., airlines are generally very committed to getting passengers home one way or another, no matter what, even if they've missed their last flight for the day. If you've got a ticket, they'll get you home.
But I have had flights like this, not where I was the only passenger, but where I was one of less than a dozen on the entire plane including first class.
And it was so nice.. and first class is extra nice.. and the stewardesses are a lot more friendly since they're a lot less stressed as well.
Although the best way to avoid screaming kids (and their parents) is to book a midweek night flight.. usually, the later it is at night, the better, or, if you can handle it, a very early morning flight.. early meaning, getting to the airport when even the Sun hasn't hit the snooze button, let alone had its morning coffee yet..
It is very quiet and relaxing, and the staff are ok with the passengers changing their seats, just as long as they are in their seat belts before landing.