Your understanding is astonishing, truly.
For an n-channel mosfet, when the gate is at source voltage (i.e., ground), the mosfet is OFF. When the gate is a few volts different than source (i.e., > +4v), the mosfet is ON.
When the gate is connected to a micro port pin (configured as an output), it will either be low (OFF) or high (ON). Therefore, it will work as expected without the pull down resistor.
With the gate floating (i.e., not connected anywhere), touching it may bring it positive enough to turn the mosfet on. If the gate is floating (something you don't want), the pull down resistor keeps the mosfet off. Twenty posts ago, you started with "the mosfet won't turn on", so may this is ¿progress? Ten minutes of research to understand all of this would have saved 4 days of frustration.