Simple Mosfet driver
Yes, the transistor-resistor driver is helping.

A quick look at the datasheet of the MOSFET show Qin = 38nC and Cin = 860pF.

With I=2mA direct drive, switch time is t=Q/I=19us. With your driver, pulling up, the RC time constant is t=R*C=40ns (way faster), and pulling down, that BJT must have some current gain. So yes, it appears to help quite a bit.

That is the "back of the envelope" approximation. You can simulate it very easily to check for other "2nd order" effects.