The clicking is likely coming from the TDCM which is timing out (approx 1 second) from the time the key is turned to RUN until the relay opens in the TDCM if no operator is sensed to be in the seat by the seat safety switch.
The tractor should still crank and start even if the seat is empty IF the ground speed lever is in the neutral slot AND the PTO switches are all off... Here is the wiring excerpt:
Verify that you have 12 volts at the key switch terminal S2 in the RUN position -- that will confirm all the starting interlock switches (your tractor is not new enough to have the parking brake switch.) If no voltage is present there check for good connections on the fuse holders -- your tractor is the older model with the glass cylinder fuses and plastic tube/spring contact holders and those are notoriously unreliable.
After that, if you do not get starter motor cranking even when voltage is present on S2 when you turn the key to START, then it may be the solenoid itself, the starter motor or its brushes, or even a bad key switch at the S2 to S1 connections or the mating connector at the key switch on the harness...
Here is a "roadmap" of where components are physically located on the 316 Onan and 318 tractors:
I was having some difficulty following your descriptions in the first post of this thread due to missing punctuation, but I think that the five wire "cube" mounted near the starter is a relay providing the "starter improvement kit" function. If that is the case, then it should be wired like this diagram:
...and physically look like this if it was done using the Deere AM107421 pre-wired kit.
Hope this helps...I will send you some information too large to post here via e-mail.