Well, several things. First of all, not sure if your unit has a separate solenoid or one built into the engine, I am going to assume separate.
If it is a 3 wire solenoid, meaning 2 large hot cables on the top posts and 1 engage wire in the form of a spade connector, you could have a bad ground. It grounds through the metal that mounts it to the tractor frame. If the bolts are loose or rusted I'd clean them and the frame below the solenoid and then remount and try again.
If it's a 4 wire solenoid meaning it has 2 top posts for starter positive cables, and a negative and engage positive wire in the form of spades, check to see that the ground wire isn't broken or loose.
I would have a helper sit on the seat and try to start the mower with you.
If you remove the trigger 12 volt cable (smaller wire) , whether it be orange, yellow, whatever, and it tests for 12v positive as someone turns the key to start, you know your good there.
Plug the trigger wire back in and test the voltage again. Turn the key to start and if it shows 12 volts, move your tester to the starter side positive terminal on top, which should have no voltage unless attempting to start. If that pole shows positive for 12volts then your starter is either bad, or stuck, or something along those lines. Make sure the terminal on the bottom of the starter (where the larger red cable connects) is also showing 12v when holding the key to start. This will rule out the solenoid itself I would think.
Here is a quick vid on how to replace with a universal version if the solenoid appears to be the issue. He also shows how to jump the terminals to make sure the starter itself is functioning. If you connect the 2 positive terminals with a jumper or otherwise, and the engine turns over, then you know your engine and starter are fine.
https://www.youtube.com/watch?v=2NzaU9jDeKs
I can't see why the solenoid trigger would show zero voltage for any reason honestly, if it's being sent 12v off the solenoid, it should still show 12 on it.