Re: Simplicity Regent starter problem
I'm not sure if your engine is one that uses a compression release activated by the cam. If it is, check your valve clearances carefully. Too much clearance negates the compression release, and gives the symptoms you describe. The cranking better with throttle closed is also an indication of an inactive compression release. Since less air gets into the cylinder with the throttle closed, there is less to compress, putting less load on the starter.
I would also suspect a weak battery, but since you mention the cables being warm, the battery probably puts out okay. It's also possible that the starter is weak.
Let us know what you find...
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2006 Conquest, 54" deck, Vanguard engine