As a UA1K I fly on business and with my wife and 18-month old quite frequently. Like everyone else when I am flying, whether in 1st or coach, I don't think that sitting next to a screaming baby is all that much fun (even when its your own!).
That being said I don't see any reason why a family should not be allowed to have their child up front. As a perosnl matter I have decided not to bring my daughter up front because I think it is much easier to seat her in coach where we can have three seats across or even an empty row (we tend to fly SFO-EWR alot on 757's and 320's). Maybe when she is older we can move up front but for now coach is perfectly acceptable.
I also wanted to add a comment on the diaper question. Unfortunately the lavs on 757's and others are not big enough to lay a child down on the seat to change her (320's and 747's have pull down changing tables which are great). Consequently we have to change the child in the seats and until UA puts changing tables on all its planes I don't see another alternative