Absolutely sure. The first thing I did when I got home after a day of running a heater in the tractor shed was to try it again to rule out water freezing in a fuel line. It had the same exact symptoms, starts every time, then immediately stops. I then replaced the solenoid and it ran normally.
I would note that, not being familiar with working on these much, it took me about an hour. I had to remove the entire cowl and hood to get to it. Then, the allen screws were VERY tight, not so easy to get to and I was VERY concerned with stripping the hex. They were rusty inside the hex, had to be cleaned out and the allen key had to be tapped in with a punch to make sure it was fully seated. Then, vice grips were needed to get enough leverage to break the threads free.
Hope this helps others. (Our local Kubota dealer service manager was insistent that if the solenoid was bad or it was not getting power, the tractor would not fire, AT ALL. Obviously, he was wrong about that. Also, he was absolutely convinced it was a saturated, frozen air filter (wrong, again).
THANKS AGAIN TO EVERYONE! (Saved me a 2 weeks of downtime with it going to the dealer and at least $600 to $700. Part was $137.)
DOC