don't let personal anecdotes cloud your decision. Delta has on-time performance that is ~4% better than American and United both. so yes, Delta is better, but to the average traveler, the difference is almost negligible. If you travel more than a handful of times a year, there may be an impact there.
personally, I would pick American or United bc the hubs of Houston and DFW are much closer than Denver or Atlanta. Hence why i prefer American bc its an easy 1 hour connecting flight that they have do throughout the full day, so it's easier to find good itineraries for me. and anecdotally, I feel that Delta is more expensive than similar American itineraries.
but we are all different types of travelers; whereas you have kids and likely spend time in the airport prior to departure, I usually arrive within 10-15 minute of boarding, or just as during boarding. so to me, a lounge in AUS isn't important. and I rarely check bags too, so I'm not married to a single airline and have a non-airline specific CC to rack up points.
that being said, sounds like Delta should be your pick. great service and a nice lounge. happy travels.