Had the exact same thing happen to me 2 weeks ago. I flipped the circuit breaker off and on a few times, it's located outside in a box but still, and it kicked on. Last week, found a local AC guy that seemed legit and he went through my 22 year old system. The capacitor was a little weak, but still pulled duty. He explained to me that the capacitor on these units, a 97 Lennox, 4 ton, built date of '97, not only use the capacitor to fire up the compressor, but it also helps regulate the compressor so if capacitor is not running 100%, could eventually do damage to the compressor. Had it replaced, only about 60 bucks.
Another thing I recently did and it MADE A HUUUGGGEEEE DIFFERENCE, was I installed an electric attic fan when I got my roof replaced. Set the thermostat to 98F. I have also had a remote temperature reading sensor in the attic. I can remember many years looking at the in the summer and seeing 125F many times. With this new attic fan, I've yet to see it go past 105F. I'm sure the cost of electricity running that attic fan offsets the cost of running/straining the AC unit. Wish I'd done that years ago.