Posting the Sears 917.xxxxxx number allows one to look up your specific tractor.

Basically-
PTO Bad or unplugged.
PTO Switch bad or plug not making good contact.
Possibly an Operator Presence Relay (OPR) bad or unplugged.
Since we don't know your tractor number, one can't look at the proper schematic.
At least "some GT5000's use an OPR to complete the ground circuit to the PTO.

When I use the term "unplugged", it may be a bad connection on one or more of the contacts, not unplugged in the conventional sense.

Yep, that all important technical model number will be key in getting some facts about the wiring scheme.
Look under the seat on a decal for that number and post back.
And another possibility is that your tractor might have a PTO clutch that has an adjustable air gap. If so, at some point in time the air gap will need to be adjusted to enable the clutch to engage. Wear of the clutch surfaces over time increases the air gap to the point it can become too wide for the magnetic field to "reach out and grab it".

"... the engine was difficult to start (had to jump the battery)..."

Although the tractor will start and run with a jump from another battery the electric PTO will not engage without a good battery on board the tractor. Have the battery load tested.

When you hear hoofbeats think horses... not zebras.

Not charging, PTO will not engage and will kill the engine..

Your alternator has lost its internal ground to the rectifier (voltage regulator). To put it simply. They can not communicate with one another.

Try this 99.9% high dollar savings tip.

You will need a piece of 14 or 12 gage wire with about 1/4" of bare wire exposed on each end. 16 or so inches long will be good for testing.

Start the engine. Adjust to about 1/2 throttle.

Take the piece of wire and hold one end of it to one of the screws that holds the REC to the side of the engine (left side with purple and white wires plugged to it).

Ground the other to a bare spot on the frame. DO NOT USE THE BATTERY GROUND for testing.

Watch the AMP meter.

If it goes completely to the right + (pos charge 30 amps or so) You have found your problem.

Shut the engine off.

Permanently attach one end of the wire to the Rectifier and ground the other to the frame.

Your charging system components will now communicate to each other and charge like it should.

Once your battery returns to full charge the AMP meter will be near -0+ while running and jump toward the + side for a few seconds right after starting or engaging the PTO or lights..

You can thank me later for saving you a ton of money on useless expensive parts, labor and work.

What decade are you working in?
What's a REC?

Sounds like you are trying to polarize a generator which is a BAD thing to do on an alternator!

REC is simply short for Rectifier/Regulator. They no longer call them a "Voltage Regulator".

{{gwi:317727}}

I have not saw a generator on a YT or Gt mowers in years. I am telling him about a newer Kohler engine set up.

I have used a piece of wire to connect to one of the Rectifier screws that hold it to the engine cover and the mowers frame for years. And have used the method on many Kohler engines to bring the charging system back to life even after the owners have spent a wad of money at authorized dealers only to have the problem return within a few days to a few months.

All it does is reconnect the communication between the REC and Alternator.

For example.

Take a 25 Amp System that produces 14.6 volts. If the engine is running and only needs 5 amps that is all the Rectifier will send to the needed components.. Engage the PTO (more Amperage needed) and the rectifier should allow the amps needed to pass through the Rectifier and keep everything running smoothly and send the surplus amps to the battery. If it doesn't the engine will shut down as soon as the battery goes dead.

I have used it on my own Husqvarna GTH2754 since about 2009 and have yet to experience the first problem.

Sounds like he is temporarily grounding the regulator mounting screw. Rectifier if you will. My high dollar saving tip is to either unplug the regulator or remove the battery ground momentarily before diagnosing (some) Kohler charging systems. That is all...

Whether the charging system is working or not is moot if the battery is bad.

With one wrench the battery is out and a trip to Autozone and a load test answers the battery question. With a definitive answer to the battery question the OP is at least one step closer to resolving the problem.

Since the tractor had to be jumped to start my money is on a bad battery.

Are we allowed to say WTF?

OK. WTF.

Not aware of any Purple/White wires on a Craftsman.

I guess it's a moot point since the OP doesn't seem interested in responding.

Yea, but we all learned about polarizing diodes.

The OP is probably not responding because of a fire at his residence.

{{gwi:317728}}

There you go. Ran a wire from the Rectifier mounting screw to the frame. Been that way for five or six years and have yet to replace a battery, charge a dead battery or replace anything related to the charging system.

Say it again. I have repaired many Kohler engine charging system with this simple method and have yet to hear of anyone having any problems after the fix.

PS: The blue battery wire from the rectifier used to be Purple until it faded.

Since it's a GT 5000 it could also be a Briggs.

I am sure that I read... "(had to jump the battery), the 25hp Kohler,,,"

If the charging system is operating correctly and supplying enough AMPs. Once started. You should be able to disconnect the battery and everything on the mower still work as it should.