Take the starter to a Auto Zone if one is close by and they can test it and determine if it is bad and in need of replacement. If the replacement route is pursued, you might consider contacting this company. KUBOTA 16241-64010 THRU 19269-63013 ALTERNATORS AND STARTERS BY OEM #. You will probably suffer a severe case of "Starter Shock" if you purchase at the Kubota Dealers.
Starter still trying to start the tractor without the key even in the ignition sounds like a wiring issue to me. The big heavy starter wire is normally hooked straight to the battery. And the smaller wire is the solenoid wire that momentarily gets 12v when trying to start. (thats a typical start curcuit and I dont know for sure if thats how the L-series are). I'd have to look at mine. But I'd be unhooking that small wire and see if it has 12v on it at all times. if it does, there is nothing wrong with the starter.