Welcome to LAX. As a long-term AA plat/EXP, I think AA would serve you well. I'm less familiar with the other contenders. For west coast trips, Alaska/Horizon miles can go to your AA account/status.

LAX deservedly gets a bad rap for its outdated physical plant and lousy transit connections. A long drive to LAX with unpredictable freeway traffic can be frustrating and lead to lots of wasted time. But if you live nearby and learn a few time-saving tricks, LAX can be an efficient, pleasant home airport.

Ex: The upper level roadway (departures) is packed with passenger dropoffs in the morning and the lower level is packed with arrivals at certain evening hours. If you're being picked up or dropped off at these times, avoid the crowded roadway and take an escalator or elevator to your level.

The AA elite security line is fairly well arranged (after years of mess) and your wait will usually be minimal.

LAX Admirals Club is a good one; joining can be a worth the $$ if you have lots of pre-departure waits at LAX.

There's a semi-helpful thread in the California/LA sub-forum about LAX parking options.

For short trips, you may be able to justify parking in the structure right across from your terminal. It's pricey, but quick to/from the gate.