The current drawn by the starter motor depends on how well that solenoid pulls in and seats the contacts. When the starter is new a poor connection at the spade connector would be less obvious. Over time the faces of the contacts burn, you get a poorer connection, and the starter motor can't get the current it needs when the spade connector has a poor connection. Clean the connection, the solenoid gets a little more current and it seats the contacts enough for the motor to start.
I'm almost willing to bet there's been a lot of starters replaced due to this that were still perfectly fine.