Firstly the problem is not in the solenoid.
The only thing the solenoid does is to turn the power to the starter motor on & off the rest of the time it just sits there going rusty.
If the fuse is blown then the ignition switch gets no power FROM THE BATTERY thus it can not trip the solenoid to start the engine.
When running the ALTERNATOR puts power into the ignition switch thus things that won't work when the engine is not running suddenly start to work.
If there is no power from the fuse then the fuse is bad or the fuse housing is bad.
I don't know what you have read, but what you should have been reading was the JD Technical manual then you would not have just bought a $ 30 solenoid that you did not need to replace. ( 1/2 the cost of the manual )
From the solenoid hot terminal a wire runs through the fuse then to the B ( battery ) terminal on the ignition switch.
From there it goes to the S ( Start ) terminal & to the A ( alternator ) terminal & to the PTO switch.
If there is no electricity coming in then there can be no electricity going out.
Starting circuit is totally independent of the running circuit and the PTO circuit
STARTING
Fuse to B
B to S
S to PTO
PTO to Brake
Brake to Solenoid
There are diodes all over the place on a JD so God only knows haw the circuits work when you are back powering them through the alternator.