I would think one global airline that appears to have come through covid with a very good reputation (in general) is QR. They never stopped operating many flights - including to/from Oz and were often the only options to get people places when many others had cut back a lot or fully.
Yes, I know, THAT deplorable issue occured that is genuinely upsetting and horrible (and I am not female so can't pretend to comprehend fully) and let's not forget that, though I also tend to feel that was mostly a Qatari government action and not QR per se (though they are both one and the same in many respects too.. so grain of salt/). At any rate I fully understand and respect various folks not wanting to go near the carrier with a barge pole.
Those issues aside, hopefully very very isolated, it would be one brand that has stood up well.
Back OT to QF though.. they still have a huge share just because of their size in the dom market, loyalty tie ups, and the fail on VA1 and VA2 still being relatively weak(but getting much better it seems).. Rex .. floundering I reckon, and Bonza still a no show(and who knows how that will go anyway).. so QF would have to have grounded the whole airline (oh wait...) to lose a huge amount of market share in the domestic space... it's just how it is (see Air Canada as example).
Internationally though QF's market share is under 40% IIRC and dropping. look at carriers like QR and SQ who are upping capacity and flights to major AU cities (and even smaller ones, like return to CNS with the Max).. I noticed at MEL the other day there were 4 SIN flights on SQ - 2 of them in the space of 3 hours. That is very impressive. Mother QF only the one (plus JQ).
Add in carriers like UA also ramping up (SFO-BNE, SFO-MEL, SFO/LAX-SYD) and others, there's a whole lot of other options for international than QF and JQ - often with better product, or better connections to where folks want to go.
I do agree the general issues over the past few years, but SPECIALLY the past six-nine months post opening up has exposed more and more ongoing issues to the general populations (eg: issues with call centres, flight credits, etc) and this has definitely tarnished the brand (not to mention your CEO blaming customers and all of these PR faux pas)