So the 15$ part is not worth anymore of my time and since the mower is old replacing the part is a good idea for myself. Before installing the new solenoid, I tested it just like I did the new key switch. You never know and that test will save time should it be a dud.
So to test the new solenoid I got my NOCO XL jump pack and clamped the black clamp to the install bracket on the solenoid to ground it and touched the red clamp to the lower pole on the solenoid and I got a big click. I set up my electrical tester for a capacitor test and put my red clamp back on the lower solenoid post and quickly checked the two top posts and it beeped and then quickly unhooked my positive jumper off the bracket. That simply quick test means I can use the new solenoid with confidence.
When I replaced the battery at the beginning of the season I freshened up my ground on the battery and I will do the same thing when I remove the old solenoid, ensuing it has a good ground mating surface. The base is the ground on a 3 pole solenoid. I have new safety switches on order and until they arrive I will keep those jumped. It should crank up after the new solenoid install. The wiring was not stretched or damaged so I doubt anything is wrong with that. The harness switch plug had plenty of dirt and such packed in the plug and I did my best to clean that up during the switch install so that hopefully will not be a problem