That sounds like a bad connector/ground.
If you measure voltage at the plug, but the (tested good) motor doesn't spin, look for a corroded/damaged connector.
Voltage drop across resistance is a function of current flow. So if you have a high-resistance (corroded/damaged) connection, you will see full voltage with a multimeter because that only flows a milliamp or so. Then when you put the blower motor on and try to draw 10+ amps, the voltage drops to zero.
