Small Engines (Lawn Mowers, etc.): Engine clicks, but will not start, craftsman riding lawn mower, briggs stratton engine

Question
Craftsman Riding Lawn Mower
13.5 Horsepower; Briggs & Stratton Engine
Model#: 917.252501; May 1995

Cut the yard one weekend, and then next weekend the engine won't turn over.  I turn the key, hear a click (solenoid), but the starter won't spin and engage the flywheel;

The manual recommends checking the following, and here is what I found/did:

1) weak or dead battery; > no, fully charged;
2) corroded batery terminals; > no, very clean;
3) faulty solenoid; > no, replaced it with a new part, but got same result;
4) faulty starter; > not sure; bypassed it with a jumper cable, and I can at least get it to spin, but it will not engage the flywheel;
5) loose or damaged wiring; I have no idea...

a) The clutch/brake pedal safety switch works;
b) The blade engagement safety switch works;
c) Not sure about the seat operator presence safety switch; seat up or seat down, I still get the click.

Please help.  This is making me nuts.  I have already paid someone else to cut the yard once, and I am reluctant to take it in to the local repair shop due to past experience.

Kind Regards,

Rick King
Pleasantville NY

Answer
Hi Rick,

You are on the right track and have done most of the inspecting work already.  You have isolated the problem to the starter drive.  If the starter was jumped, it should not only turn over but engage the engine's flywheel and crank the engine.  The bendix or worm gear on the starter may be broken or dirty and in need of cleaning and a coat of a teflon type spray lube.  
A weak battery (even though charged) will also cause the starter to not engage, so test the battery or jump the starter using another vehicle.  
Now...Turning the key should allow the starter to engage, just like when it is jumped.  Once you get the starter to engage there may be another problem entirely, unless the problem goes back to a bad battery.

Let me know how you make out.
Michael