The start capacitor is not really acting as a capacitor, it actually shifts the phase of the power supply slightly to make the machine run more efficiently. I don't know why some manufacturers and technicians call it a start cap; it isn't powerful enough to make a difference and can actually be eliminated entirely.
Frigidaire (original maker of this machine) properly calls it a run cap I think. Last week I saw a machine that had been running for 5 years without one wired in at all. There were 2 sitting in the cavity but neither was wired in. The compressor has an internal separate start and run winding that handle the additional power needed to start the compressor.

Your overload may be getting sensitive or your compressor may actually be drawing too many amps. There's no way to be sure until it fails to start and you swap parts or you can test it properly. You can check amp draw with an inexpensive clamp style or non contact multimeter.

I don't believe your compressor electrical parts are available at this point, if you can't track them down you can try a hard start kit like this:
Pro TJ90RCO410 3 in 1 Relay Combo Kit - AppliancePartsPros.com

Assuming the compressor is working normally inside that can replace the discontinued relay/overload and may get more life out of the unit if the compressor really is harder to start than the original overload likes.