AC makes great points with excess current draw clearly causing the red lead to burn out.
I would certainly test the capacitor and this is where the "death head" comes up.
Capacitors store charge and a good capacitor can store a REAL hefty charge for several days (or longer). Before you do anything, remove the AC power and short the capacitor terminals to each other. You should see a big spark and the snap of a high current pulse.
When uou can't find the exact capacitance value, get two capactors and put them in parallel. This placement allows you to simply add capacitances. The voltage rating is not critical so long it is ABOVE the stated voltage on the original capacitor. In this application, it is NOT critical if the total capacitance is a bit larger, the key is not to have it be too small.
Now the interesting way to test . . . . yeah, not with small kids or animals around.
Disconnet the leads to the compressor and put main voltage (120 240, etc) on the unit. Most times this is via a circuit breaker. Just a minute or two is sufficient - then turn off power.
Now get an insulated screwdriver and short the terminals of the capacitor together. The louder the snap, and the vigorous the light pulse indicates the amount of stored energy. A VERY quick and dirty test, but if there is a weak spark, thern you know the capacitor is likely bad.
In enginering grad school, we would take a much smaller capacitor, wrap the leads around the non-metalic capacitor body (leads not touching) charge it up - and then toss it to someone who just walked in to "catch".
The auditory responses were quite impressive . . . . .
Steve