A battery that reads 12+ volts does not mean that it can turn over the engine, or even pick up the solenoid. To do this you need current. An old battery that shows 12+ volts, may have not enough current in it to light a light bulb. A brand new battery, right off the shelf, may not have enough current.
Having to jump a battery from a bigger battery to start, to me, indicates that the tractor battery does not have adequate current in it to do its job. The CCA rating (cold cranking amps) is insufficient. The problem could be compounded by poor/dirty connections, inadequate water level in the battery, cold battery, cold motor oil, improper weight oil, points/plug not set properly, eletrical parts binding, etc., etc. Could even be valve clearances that are out of whack. Could even be improper sized wire used during a re-wire.
The video of how to 'test' a solenoid is not 100% perfect. True, it does show if the solenoid picks up and completes the high current path, but it does not show how well the high current path conducts during the starting process. The go/no go test passes a very small current thru the 'high current' contacts. During a start operation, you should be drawing 100 amps or more to the starter. The 'high current' contact faces inside the solenoid may be badly pitted and corroded internally, allowing only a few amps to pass thru. I have seen highly polished high voltage/current contacts in 345kv. circuit breakers that show perfect conductivity at low voltage, but refuse to pass any current at 345kv.