Has the igniter ever been changed? This is the chip that tells the coil when to fire. That and knowing that the choke is working as it should come to mind. Here is a link that might be useful: KAW manuals

JD and Kawasaki have partnered for quite a while ion the upmarket JD garden class tractors. I believe the reference to the iTorque is to recognize the latest class of Kawasaki engines which meet the latest EPA requirements in 2011. If you go the the JD web site anf punch up the operator manual for the 2011 X500 it clearly shows: Engine - iTorque Power System X500, X530 and X534 Engine Make: Kawasaki Model Number: FS730V Bore 78 mm (3.07 in.) Stroke 76 mm (2.99 in.) Displacement 726 cm3 (44.3 cu in.) Cylinders V-Twin Cycle Four Cooling Air Cooled You can Google the model # FS730V and look up anything you want to know about the engine.

Sure sounds like a fuel supply issue to me, but kinda baffled by your "throttle back" description for remedy. I would pull the fuel line downstream of pump and filter, but upstream of card and observe flow. If that seems normal, I would pull fuel bowl on bottom of carb and check float is not sticking. While there check for dirt particles. When it starts to die if possible, shoot some carb cleaner directly in carb. If accelerates, then definitely fuel supply issue, and carb clean/kit would be my next move. Hope it helps

If......gasoline is in the crankcase, it's not that a vacuum is created when the air filter is in place drawing out the gas/oil mix. Gasoline in the lube oil begins to vaporize out of the oil ( as soon as the piston and cylinder begin to warm, which is almost instantly upon start up) and creates its own specific pressure as it does so. This specific pressure will vent of its own volition without any partial vacuum being applied. What is actually happening is that when the air filter is installed, is that the venting gasoline vapors are confined inside the confined space created by the air filter and since the gases are coming out of the crankcase vent under pressure..........mostly what enters the carb is a fuel rich mixture with very little air because with a pressure fed source inside the air filter, there is no way for sufficient air to be drawn through the filter media to make a proper air/fuel ratio. Another phenomenon that may plague engine air filters during cold ambient air temps is air filter icing. In humid conditions with ambient air temps near or below the freezing point of water, ice can form on the filter media and cause drastic clogging. This is why most engine driven snow blowers/throwers do not feature an air filter. Both the fine mist of snow particles and the high water vapor content (from snow contacting engine and exhaust surfaces) produce conditions which are perfect for air filter icing.