First step is always to go to the web site and see what information is there for free.
So go to the Ariens web page.
In the top right corner in a fine font is the words Product support, click on them
In the next window click on the Manuals link then on the next window pop those model & serial numbers in.
You find the owners manual, parts list & service manual all there for free ( bargin ).
In the front section is the wiring diagram which while being a little confusing should point you in the correct direction.
In the electrical section you will see that the mowers with the Briggs or Tecumseh engines use a 4 pole solenoid.
The green wire goes to the S terminal on the key switch so should show + 12V when the key is in the start position.
The other wire ( yellow) provides the ground which completes the circuit so the solenoid can trip & supply the starter with high current 12V DC .
Studying the wiring diagram reveals that the yellow wire bridges at a lug and one side goes to the PTO switch then to the Neutral switch on the gearbox then to the seat switch and then to ground via the brown wire.
All 3 switches have to be in the closed position for the ground to get to the solenoid.
SO check the Yellow wire for ground
The seat, PTO & Neutral switch generally have a 2 blade plug so they can be replaced .
If that is the case make 3 short jumpers with a spade on each end and use them to bridge the plug thus bypassing the switch.
Continue till the engine cranks .
If it does not crank then test for a ground presence as the brown or black wire could have broken.
The kawasaki egine is very similar except they use a relay.