To me, that sounds more like a belt slipping, with the burnt rubber smell.

I'd agree except that the only belt that gets hot is the main drive belt from the clutch to the deck drive pulley. Belt appears to be in good shape and tension looks right on it. Couldn't a slipping clutch be heating up and causing that belt to get hot?

IF a malfunctioning clutch was getting hot enough to start burning the belt, I would expect the clutch to be dead first.
Anything is possible.........I guess.

I've checked both of the deck belts and they are cold. All spindles are slightly warm. Only belt that is more than slightly warm is the belt from the clutch to the drive pulley on the deck. I'm not positive it's a 'hot rubber' smell, but definitely something hot. Is it possible the clutch is slipping, getting hot, and producing the smell? I have watched the clutch belt when I hear the 'chirping' noise and it does not appear to be slipping. As a sideline, I've had problems activating the clutch this past week. Had to flip the switch on and off a few times to get it to work. I noted from other posters on this forum that is usually an early symptom of clutch problems. Thanks.

Do you know if the mowing system on this tractor has a "blade brake" built into the PTO clutch?
The blades must stop within about 3 seconds (federal mandate) after the blade drive is disengaged.
Some OEMs build the blade brake into the deck pulleys configuration while other OEMs are using a PTO clutch that has the blade brake function built into it.
If your PTO clutch has a blade brake built into it, there are a couple of things I can think of that "might be" causing the symptoms you are seeing.
1, the PTO clutch is defective.
2, the PTO switch (or other items in the circuit) has become worn to the point that the internal contacts are "chattering" (for want of a better term) and so the clutch is attempting to disengage and re-engage rapidly (chirping?)

The fact that you are having trouble getting the PTO to engage now could be pointing to either a defective PTO clutch, or a defective PTO switch.
Beware that the second scenario could be caused by ANYTHING in the PTO clutch circuit that has (or makes) a poor connection between the PTO switch and the clutch itself. This means that a loose fitting wiring connector (maybe at the clutch itself) anywhere in the wiring harness might cause the kind of electrical "chatter" I described.
Some PTO circuits employ the PTO switch itself to handle the electrical load of the PTO clutch. Some other machines use the PTO switch only to operate a relay that carries the actual current load to the clutch. If a relay is used, scenario #2 could also be caused by a loose connection in the relay or related circuitry.
Eliminating any switches or relays to isolate the problem could require hooking a temporary jumper wire connected to the PTO clutch wire port on the clutch (after unplugging the OEM wire). If one were to opt for this as a test it would be necessary connect the wire to the clutch first and then connect the other end to a battery positive point after starting the engine. If a 12VDC power plug is featured on the dash board, that would be an ideal place to connect the top end of the jumper wire.
If no 12VDC power plug is featured, I suggest that a jumper wire be made up that includes a 20 Amp fuse along with a simple single pole toggle switch to be connected to the battery + post. Tape any exposed terminals on the switch.
In lieu of an actual toggle switch a simple "male/female" wire connector pair could be used to connect the jumper wire together for the test, though it would a bit more cumbersome to use than a flip switch.

Before I resorted to doing the jumper wire test I would certainly do a thorough visual inspection of every wire connector from the PTO switch all the way to the PTO clutch to be sure that all those are secure and free of any corrosion and that there are not any signs of burning/melting of the metal terminals or the connector body.

Good info. In addition, it is a good idea to verify that the clutch bolt is also secure on these. One would assume no powerflo nor any oil leak?

Have your battery load tested. A marginal battery can cause some odd PTO behavior. Check battery cable connections on both ends of both pos and neg cables.

Dirt, dust, and grass in the PTO can cause some odd smells too.

There have been (rare) occurrences reported of the clutch bolt unscrewing and/or snapping off on JD Select Series lawn and garden tractors so check that bolt.

I can't find an adjustment procedure or specification anywhere in my X500 tech manual so I don't believe there is a gap adjustment for that model PTO.

"Hot rubber smell" and "imminent electrical failure smell" can be VERY similar.

If you have a marginal battery or a loose or dirty electrical connection or bad ground the resistance at the PTO will rise and the temperature along with it.

Could be your PTO is on its way out but I'd check the things mentioned in this thread BEFORE I bought a PTO.

Okay, battery failed the load test so I replaced - 6 years old, about time anyway. However, I still have the problem - but I think I've isolated it to the deck idler pulley - the one attached to the pivot - as the service manual describes it. When I run the deck a couple of minutes, the chirping, then squealing starts. I shut it off and touch all pulleys and that one is hot enough to burn the fingers - so I'm guessing it is the culprit!

Good guess... you gotta lube that idler and the pivot. It gets overlooked because there's no Zerk to call attention to it.

Take it apart and see if it's scarred or galled. Replace or repair and lube.

Oh, I'm sure it's a replace, not a repair. Once I figured out it was the problem, I watched it and it was hopping up and down about a quarter inch. I'm sure the bearing is gone. Service manual says to grease the pivot but doesn't mention the idler. Thought it was sealed. How do you grease it without a zerk? Can I replace it without pulling the deck? Thanks...

Much easier to pull the deck and it only takes minutes to drop.

I use Bel-Ray 6in1. It's GREAT stuff sold at motorcycle shops. I dribble a little around the shaft every so often and it does the trick.

Great stuff for contact lubing anything.