Well, with the check valve removed, that rules another item out of the picture.
That one sounds like a Run Capacitor.
Normally the starts will have ratings in MFD.
If you have a multi tester with a CAP settling, yours can be tested before getting one to be sure.
If you do not have a tester capable, maybe there is an HVAC contractor somewhere in the area.
I am sure if you stopped by, they would be accommodating to quickly test your capacitor.
They could also help supply one much faster if it is indeed bad. Since they have quick access.
There may be some building supply stores close by as well.
Would be nice to verify before getting one and still have an issue.
The motor could have a bad set of windings.
To test the motor, A local motor shop could assist or a pool supply store many times has these parts for pool motors. Fairly common.
You have the numbers needed and also can carry it with you.
the key is the 30 UF 370 volt
You can use any the same physical or small size at higher voltage also.
30 UF must be used but you can use a 440 volt as well as a 370 volt.
Many sites do not list the actual dimensions, because they acquire from various resources and the size changes from time to time.
In hand is the easiest way for sure.1 1/2" x 2 5/8"- CLICK HERE