My experience (and only that) is a bit mixed as a 6+ year DM (prior to Covid, when things became a bit less predictable, understandably) and a few before as PM. Summarily:
- bags gone missing (including once having bag sent to MEX and very clearly used for suspicious purposes), zero response. I don't think they care, once the bags make it to you.
- complaints about service issues (e.g. FA yelling at other pax, ignoring all of FC for a trans-con flight): usually handled quickly, always thrown a few thousand sky miles. This seems to be something they pay attention.
- compliments/good things: within 10 days, a follow-up email saying thank you
- anything requiring a refund (e.g. downgraded class flown): either VERY fast or VERY VERY slow, it's a bit 50/50 on this.
Everything right now is a mess, so I wouldn't expect much, but that's air traveling in summer 2022. TL/dr: pick your battles and know that the grass may look greener, but it probably isn't.