Fully charged battery. no response when key is turned to start

I like to start by making sure the starter gear not locked into the flywheel , this happens a lot on briggs take your hand turn the flywheel screen counter clock wise it will walk the gear down , you should be able to turn eng by hand now if not gas leaked into eng .
or eng is locked up.
Next I like to jump the starter solenoid to find it follow the heavy red wire from the battery it will led to it, there are 2 kinds 4 post and 3 post looks like you have 4 to test a 4 post unplug both small wires from the spades , you will need to small pices of wire hook one small wire to spade and ground it to frame of tractor now take the other wire hook it to the other spade and touch it to large stud that leads to the battery make sure you have parking brake set it may crank over, this by passes all safety switches and starter switch if it cranks over battrey good starters good solenoid is good .
safty switch on clutch pedal of mower deck engage switch not seat switch .
IF it does not crank over use a heavy wire or pliers jump large stud to large stud if cranks over solenoid is bad .

Sounds like a safety switch concern. Most mowers need the brake or clutch pedal depressed or the park brake engaged for power to go to the starter. Could also be the seat switch. My John Deere LT160 needs the brake pedal depressed or I can lock the park brake on and start it without being in the seat. Hope this gives you some ideas.