The issue is the run cap keeps blowing. Sounds like either the potential relay/centrifugal switch is stuck open or start cap has failed so it should just buzz until the run cap blows up, which is the symptom. The run cap keeps blowing. So removing it should cause the motor to start running then stall if if everything is normal or not run if the start circuit branch is disabled without blowing out another capacitor. Lots of ways to troubleshoot this but the only moving part is the switch.
I have fixed many stuck switches, relays, or blown start capacitors. Seldom does it take out the run capacitor as well. If the motor can develop enough torque to get rolling the run capacitor will accelerate to speed, just not as fast as with working start capacitors. Once up to speed you have same thing you normally have in the circuit. This could possibly lead to earlier failure of the run capacitor, but I doubt is what is going on with OP as he replaced original motor with a different one and is still having some problems.
If you do lose run capacitor but still have start capactitors and proper cut out of them - it will start somewhat normally but will have less than normal running torque.
We don't know if second motor lost capacitor immediately or if it took some time to develop. Immediately probably means defect or serious voltage issues.
Overloaded motor (mechanical failure, poor design, improper selection of motor/pump combination) maybe could be an issue but will likely take a little time to deteriorate to failure.
