Test the voltage at the Solenoid terminals.
Pull in Winding (from start button) should have >11V when trying to start, but if Brushes are worn it may have 12V as the pull in winding earths via the starter motor.
Voltage on Starter side of Main leads (little short lead going into Starter motor, should have >10V during operation, if there is voltage (10-12v) but no go, then it's likely the brushes.
I suspect the brushes are just making contact, but not enough to pull in the Solenoid, but if you bridge the solenoid they ark and make the starter start to spin, and the vibration makes them contact better and it starts.
This would have been getting progressively worse for some time if the brushes.
If battery voltage is under 9V during cranking it will cook the Starter and you should look at replacing the battery.
Also test Voltage drop (under load) on all leads including the Earth lead, this is more practical than trying to test resistance of .0001 Ohm
..... should be no more than 0.5V loss on any one lead or junction during cranking.
Good luck