I think you're going to find lots of people with lots of different experiences, which means it's really just hit or miss. Book a non-stop and take which times best fit you schedule, and just hope that it works out.
I've personally never flown American, but that wouldn't stop me from booking a flight with them if the price/route/time was right.
We flew Delta last month and had a horrible experience - our departing flight left late (with no explanation at all), they told us on the plane that everyone changing planes was fine, 'so just sit back and enjoy the flight." WRONG! We landed at ATL with 20 minutes until our flight to MCO!!! There were about 10 of us on the Atlanta flight who were going to Disney, and we all were sprinting through the airport, having to change concourses and hope the gate number hadn't changed... UGH!!! Needless to say, all of us were steaming mad! The door to the plane closed behind us, that's how close it was.
When we flew home, our flight left late again! Thank God we had a long layover! This time there was another group of people freaking out on the plane, rushing out to run for another flight... I felt really bad for them.
Just based on that experience, I think Delta schedules the flights way too close together, and they aggravate the problem by not telling you that you're going to be late, or telling you that you won't be late when the flight is 45 minutes behind already, or anything! The onflight CS is great, but that doesn't help me when I dragging my five year old through Atlanta, trying to explain that she has to run like the wind or she might not see the Mouse.
I've flown on United more times that I can count - Denver is a United hub so you can't avoid it living here. When I've flown UAL they come over the intercom and list the departure gates for everyone changing planes, and they contact the other airlines if you are late and notify them that you're on your way. I've never been late enough on United that I had to run through an airport to change flights, but if I was I'd feel fully confident that they'd help me make that flight. I've never had an issue with their customer service, they've always been friendly and helpful at the check in counter, on flight, etc.
So, how's that for more info than you ever wanted?

Now, don't even get me started about Aer Lingus!!
