AC Unit shut off suddenly, won't turn back on

The ole 3 knob controller were usually reliable. What is around them usually sux.
The slip-on/spade connectors (wire crimps) on the back, have a bad habit of coming loose and/or burning up.
Breaker off, remove the panel's 4 lil screws from the wall.
Carefully and slowly pull the panel from the wall. Watching for loose wires and to keep wires from pulling off of the spade connects.
Make drawings or take pictures before removing any wires yourself. Putting them back on can be confusing.

Is the strainer basket clean? Good water flow? How did the pump fix itself?

Yes, a lack of water flow will cause this. A bad run capacitor among other things will also cause it, but I would verify water flow before I looked deeper.

A water flow issue will not prevent the fan from running or the pump from running either.

it’s been 20 years since I be had to deal with the old 3 knobs…. Three position: off Fan and Cool. If you don’t get the fan to run in position 2 then either the switch is bad or you re not getting power

now if the fan runs but the compressor doesn’t kick in when switch to cool, then it’s something else. Some system have manual resetable high and low pressure switch on the compressors.

upgrading to digital controls isn’t expensive and a great upgrade

Clamp or crimp?

Yep, That has been a first to look at place for our customers when all is dead.
Glad you got it fixed.

I'm guessing you have a remote pump controller (2 or more ACs using one pump).
In this case, restrictions may cause it to get a lil more warm than usual. Keep an eye on this,
Unless it was some trash on the outside of the hull that has blown away, these issues rarely clear by them selves.