The caps you found look similar enough to work. $30 to me is a low cost thing to try but not guaranteed it is the only thing wrong.
Reasons a motor starter won't pull in:
Motor starter coil burned out.
Overload is tripped (the contactor should pull in once it is reset, if just for a short time if still overloaded)
Wiring problem in the contactor coil circuit.
By pushing in the contactor manually, even with the above problems, the motor should start. That likely means you have a starting capacitor problem or you have a motor that has gone bad.
BTW, a starting capacitor problem can be tied in with the motor's centrifugal switch being stuck. At the end bell of the motor is a switch that opens once the motor spins up. If it is stuck closed it can fry the motor start capacitors. If the switch is stuck open then the starting capacitor is never put into the circuit and the motor won't start to rotate.