All electrical power to the dash goes through the fusible link, so if you have a working glow indicator, oil light, headlights, etc. then the link is OK (unless there is evidence that someone has re-wired things!)
First things first: remove both battery connectors and ensure that all posts and connectors are clean and clear of corrosion. Sometimes a connection will be sufficient for a small lighting load, but won't handle a heavy starter load.
The next likely problem is, as 85Hokie pointed out, the clutch safety switch. It should get battery voltage to one of the wires when the key is turned to "Start". It should be open (ie: no conductivity across the switch terminals) with the clutch released and closed when the clutch is depressed.
Do you have a multi-meter or 12v test light to perform tests with? Disconnect the harness connector at the switch. Ground one lead of your meter and/or test light to bare metal on the engine or the battery - terminal. Use the other lead to probe one of the harness terminals and turn the key to "Start". If the meter shows battery voltage or the test light lights up, then you have confirmed power is reaching the switch. If not, then probe the other harness terminal. It may be helpful to have an assistant hold the key in the "Start" position during these tests, so you can wiggle the probes a bit and ensure you have a good connection.
If you can't find power at one of those two terminals, then there's likely a problem between terminal 50 of the ignition switch and that connector (I assume that the replacement ignition switch is new, not used). You will have to hunt down a break in the wire. Look for places where the wire has been flexed or chaffed.
If there is power to that connector, then the next step is to check the clutch safety switch. You can set your meter to measure resistance or continuity and probe the two terminals on the safety switch side of the connector. The safety switch should show infinite resistance when released and almost 0 resistance when depressed.
Alternatively, you could use a short piece of jumper wire between the two harness side terminals and turn the ignition switch to "Start". CAUTION: This test could cause the engine to start! Make sure all fingers, toes, tools and wires are clear of potentially moving parts and that the transmission and PTO are in neutral! Make sure the throttle is all the way off! If the starter motor cranks with the safety switch by-passed like that, then the safety switch is bad.
If the switch tests good, then you need to move to the starter motor. The starter motor itself is probably OK, since it started the tractor when they jumped it straight from the battery. But mounted on top is the starter solenoid, and it may be bad or not getting power from the safety switch. There are probably 3 visible terminals on the solenoid: one to a short bus bar between the solenoid and the starter motor; one to the big cable to the battery, and one to the small black wire with a white stripe from the safety switch.
You can use your meter (set to measure voltage) and/or test light on that small wire. Again, ground one lead to bare metal or the battery - terminal and the other to the connection. Have your assistant depress the clutch pedal and turn the key to "Start". Battery voltage at that point confirms all the wiring is good. Lack of battery voltage indicates a broken wire between the safety switch and the solenoid (since everything up to and including the safety switch was previously tested).
If you have power at that point, then remove the nuts holding all 3 wires to the solenoid and ensure the connections are clean and bright.
One final test will confirm that the solenoid is bad. Reconnect the nuts holding the wires. Again, this test may start the engine, so take the precautions listed above! Connect your test light or meter to the heavy post connecting the solenoid and starter motor. Use a jumper lead or a screwdriver or a set of pliers or similar conductive metal object to bridge the lead from the battery to the small black/white lead from the safety switch. A working solenoid will make an audible "click", the light or meter will show power going to the starter motor, and the starter will crank the engine.
If the solenoid does not "click", then the solenoid coil is burnt out. If the solenoid goes "click", but the meter/light does not show power going to the starter motor, then the solenoid contacts are shot. If the meter/light shows power going to the starter motor, then the problem is in the starter motor itself, and the start at the auction was a fluke.
The Kubota parts site shows the solenoid is replaceable separately as part # 15231-63022. An intermittent starter motor is probably repairable by dissassembly, cleaning, polishing the commutator and replacing the brushes. Looks like those parts are also available separately. And as long as I am looking things up, the clutch safety switch has been superseded by part # 67111-55810.
Have fun!
