As I replied in the same question you put in the Newbie thread, AA EXP customer service is excellent. They are quite empowered and generally knowledgeable.

AA EXP status is a very long stretch starting from zero (because the status bonus adds up to such a significant portion of your 200k required), but if you have status with another US carrier they offer a match with 4 month challenge. They also have a current limited time promo to do match & challenge with status from AC, AF/KL, LH, and VS. They don't poach from other OW partners, so you have to start fresh if that's your only one available.

I have relatively little experience with other OW CS, but all would be non-elite. Just from my limited view:
BA: a few times and has been fine for what I've needed, including getting a paid business class TATL moved to AA metal due to BA cancelling mine and no other near departure times.
QR: Generally fine but have really only dealt with Privilege Club, and little experience with their general revenue ticket team.
RJ: Kind of awful, but if you get the contact of someone at a city ticket office it makes a big difference. They will not do full re-routes on other OW airlines, they insist you touch AMM on RJ metal, so makes for inefficient re-routes during cancelled RJ long hauls
QF: Awful, just purely awful but have heard better for upper tier elites.
CX: Generally good for RTW tickets, but hard to get put in touch with the OW team who handles them. For standard tickets, bad experiences only, inflexible over irregular ops re-routes and scheduling
MH: Never needed CS, but was very good at the transfer desk when I wanted to buy an upgrade on an economy ticket once
JL: Completely inflexible, so expect to follow the ticket rules exactly, and from there it's fine. Very low cancellation penalties on domestic flights if you find a better timed flight later (no changes, cancel and rebook only)
AS: Haven't needed CS on the flights I've had with them
AY: Interacted with them only twice, seem efficient and reasonable