Quick advice: Junk AA, you and your family deserve better than those stinkin' SWU's, and give those to me.

Long Advice: AA has the best frequent flyer program. No US carrier has particularly good service. Star Alliance has the best network of carriers for most situations---but perhaps not for your situation. Delta and Skyteam seem neither here nor there.

AA seems to serve most of your regular routes well enough, and I would consider staying with them.

I wouldn't fret over a one-time trip to Asia - do whatever is convenient.

As for your LAS addiction: cut down on those trips, start enjoying more miles from connections, and occasionally stray to Delta if you have to.

By all means ask United and Delta for a status match.