Being an L60 series tractor, it should have auto-throttle, where the HST pedal can be linked to the throttle. I wonder if there is an issue with the electronics. I would start by trying to shut off the Auto Throttle feature if it is on; or turn on Auto Throttle if it is not and see if it makes any difference.
It is also certainly worth looking at the basic fuel system. Do you have plenty of fuel? If you turn the key on without starting the engine, do you hear the electric pulse pump running? Is the fuel filter on tight? Is the water separator on tight? Have the filters been serviced recently? Keep in mind the L60 series has two saddle fuel tanks with a crossover. There have been issues with the crossovers on these before.
Not sure if this has anything to do with an electrical / battery issue or not. Sounds like it does not, since it restarts awhile later. However, its worth noting the L60 series - like most Kubota cab tractors - does not really have a strong enough alternator unless you pay for the optional unit. I recall plowing snow all evening with my L60 series tractor, with the heat on, wipers on, radio on, and all lighting on. I was using AutoThrottle which worked great, but since the engine RPMs stayed down much of the time, the alternator couldn't keep up. It worked great all night; I parked it in the driveway instead of the barn. Next day I went to start it up, and it was totally dead. Had to jump start it. The stock alternator on these cab tractors is 60A. The optional alternator - required if you take the rear defroster - is 130A, or more than double. That is what these machines need, and I wish I had ordered mine that way. My MX6000 cab tractor also has the CCV heater, which draws additional power.
