Engine coughs and stops after about 15 min. of cutting

here's a few sites on that engine and problems one specific to your issue.

https://duckduckgo.com/?t=ffab&q=kohler+7000+22hp.+eng+it+surges+when+not+cutting+stops+under+a+load&atb=v159-1&ia=web

Best to you

Question edited for clarity and all the typos.
Question moved to a better category than 'Garden'.

Surging is normally caused by an air leak. Check from the air filter all the way to the inlet manifold on the cylinder head. When you engage the blades the motor is under 'load'.

More information here, but says, 'not enough fuel or too much air'.

https://www.google.com/search?q=Kohler+7000+22+hp+engine+surging

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it could be many things. try to change the spark plug. the fuel lines may be close to the engine causing the fuel to boil after it gets hot.. could be a plugged fuel line or filter. the engine may have a bunch of debris in the cooling fins causing it to over heat.

Sounds like a crankcase ventilator is plugged up. It serves the same purpose as a pcv valve on automobile.

check wires and connectors--check the inline fuse --seat switch -battery look over the wiring possible wire damaged or disconnected

check the fuel filter and fuel line. it may need to be insulated to protect from heat. running on choke means it is starving for fuel . take the carburetor off and clean it real good. be sure the fuel lines are away from heat source.

check the drive belt on hydraulic pump under you seat

try and run it with the gas cap loose maybe the vent is plugged

IF connecting that one hot wire to the pto made it engage then you either have a broken/cut/bad connection/wire to the pto or the switch is bad.
What you did by using the hotwire to the pto is bypass all the wiring and the switch for the pto.
You can determine if it's the switch real quick if it only has two connections/wires to the switch. Pull the switch, remove the wires and touch them together, after you have the mower in the condition under which the pto would engage. If the pto engages it's the switch for sure.
Take all the proper safety precautions. Don't let the wires touch anything else.

First ncheck for a dirty fuel filter, if its dirty enough it will starve the engine for fuel after a few minutes, then slowly let the fuel system refill as it sits.