Absolutely put an ohm meter on the cables. From the ground at the frame to the battery terminal. From the positive at the solenoid to the battery terminal. From the small connector on the solenoid to the terminal on the key switch. You should be seeing VERY low to no ohms of resistance depending on the setting/quality of your meter.
Remember, the wiring on these tractors is of the lowest quality that will function properly when new. They do not age well and can be the source of a lot of problems.
An old farmer on another list I frequent always says: Check your grounds, check your grounds, check your grounds! This applies to ALL electrical circuits from power plants to electron microscopes. Voltage is only half the solution, you also have to have the ability to carry enough current (amps) to get the job done. Cables with internal corrosion will show high resistance (ohms) and not carry the current you need to get the job done. Doesn't matter if positive or negative cable, they both have to carry the same load.
I have seen all kinds of cables fail internally in such a way that they could not be detected by eye from the outside. Only an Ohm test would reveal the truth. Had the ground lead from the power pole to the garage fail just last year. This was a buried 220 volt A/C service entrance cable. Manifested itself in all kinds of weird ways. Even baffled the power company linemen. Split phase A/C is much more tricky to troubleshoot than the D/C on a tractor.
Think of the circuitry in terms of water.
Voltage is pressure.
Amperage is flow
Now try to fill a bucket with that in mind. Lots of pressure but a small hose will not get the job done very quickly (spin the starter).
You can have a six inch hose but with little pressure you can't get the work done. You need both pressure (voltage) and flow (amperage).
Resistance (ohms) is the same as pinching the hose. Kills both pressure and flow.
But consider the pinched hose. If you plug the open end, eventually the pressure will build up to full reading (voltage) but there won't be much flow.
Sure wish we could meet in person. It would take me less time to explain this to you than to type it out....
Bottom line, don't give up yet, you are almost there. Also "shotgunning" parts at a problem is best left to the stealerships. DYI'ers figure things out, then buy the correct parts the first time.
I bet you don't need a whole starter, Maybe a solenoid at worst and at best a good cleaning of the solenoid or replacing of a "minor" wire/connection somewhere.
As Red Green used to say: "Remember we're pulling for you, we are all in this together "