My AC trips the breaker only when the compressor starts — what's actually failing?

Agreeing from the AC side: capacitor first, every time. A tired capacitor is the single most common reason a compressor won't start cleanly — it's a small cheap part that fails in the UAE heat, and it produces exactly this "fan's fine, compressor trips it" symptom. A hard-start kit can nurse an older compressor that's getting reluctant, but only after the capacitor and contactor are ruled out. If the winding's gone to earth then yes, it's compressor/unit territory — but don't let anyone jump to that (or to a bigger breaker) before they've actually tested the capacitor µF.

Went through this and it was the capacitor — bulging on top, swapped in ten minutes, trips gone. The scary part was the first guy wanted to "just put a 30 up instead of the 20." Glad I asked here-ish first, because that would have hidden a fault behind a breaker that no longer protected the circuit. Insist they test the cheap parts before anyone touches the breaker size.

Adding a post-summer angle: after months of the compressor starting under peak load in the heat, the electrical side is worth a look right now rather than next summer. A contactor with burnt points or a weakening start capacitor often shows up first as an occasional start-up trip during the hottest weeks, then becomes a hard failure once it's fully gone. If you saw trips like that over the summer and they've eased off as it cooled, that's not it fixing itself, it's the load dropping, so now's the sensible window to have the start components checked while you're not desperate for cooling.

Related, for the other power scenario: if the AC won't come back at all after a power cut (rather than tripping on start), that's usually a protection lockout, not a fault, the anti-short-cycle delay or, on 3-phase, phase-loss protection. Worth knowing which of the two you've actually got before assuming the compressor's gone.