Couple of things to check. Number 1. Check that you have a really good battery!. Matter of fact, try jumping it to your car or other larger battery. The PTO clutch on these tractors draws a tremendous amount of juice, and if the battery is going bad, it may not have enough charge to do anything. Even if it was running, the charging circuit is not strong enough to overpower the draw from the PTO clutch. This is a REALLY common problem on newer equipment!.
2. Check all your grounds, battery connections, starter connection, etc. If everything is good, start checking continuity between different points in the system (positive lead to starter, negative lead to engine/frame, etc.) I had a Cub Cadet 1864 that we mowed with, shut down, then never could get it started. I traced the whole electrical system and finally found a break in the positive lead from the battery to the starter. The wire insulation was broken underneath the engine where the cable ran against the frame. I'm still not sure how I found it, but a new battery cable and she's been running strong for 2 years now.
3. Check that the crank case is full of oil. The new engines with "oil sentry" technology will shut down like you flipped a switch if the oil is just a little low, or if you were on a slope and the oil ran to the other side of the block. (unlikely in your case if you got it to click but something to keep in mind)
4. Check the safety switches. Just like the battery cables, if a switch is corroded it will shut you down in an instant. You can disconnect the switches (seat, pedal, reverse, etc.,) plug in your continuity tester and press the switch to see if the continuity changes. The seat switches are really bad about going out because folks hose off their tractors and the seat foam fills with water and never dries out. Then the switches corrode and fail.
Good luck,
Bill