Elevated Vgs(off) will probably turn on slightly faster, sure. More importantly, it'll turn off that much slower: the critical point is discharging Cgd, which due to Miller effect occurs as a plateau near Vgs(th) (usually higher, due to load current, but can also be lower due to common source inductance). So you're spending whole, whatever, hundreds of nanoseconds, sitting around Vgs=2-4V, trying to discharge to 0.8V and that voltage drop just doesn't draw much current. Whereas if it were pulling down to 0V, or even negative, the voltage drop would be much higher, and so current flow as well.
Why not make the emitter follower work for you? Move Q1 collector to Q2 base, now Q1 acts as an inverting switch with a pull-up to 6V. R3, R6 and C1 are deleted, R7 connects to Q2 emitter directly. Finally, add a diode across Q2, E to B, so Q1 can sink gate current when it turns on. A ~1A schottky will do (anything from BAT54 to 1N5819, B140, etc.). This can pull down to about 0.5V and up to 5.3V, with a few hundred mA in either direction.
If you don't need fast switching (only a few kHz, or just generic on-off loads), or full load capacity (up to a few amperes), you can also wire the MCU directly to it (with a modest gate resistor, 100 ohms or so). Actually with a 5V MCU this will give pretty much full capacity (Vgs(on) ~ 5V) but it'll most likely still work even with 3.3V MCUs.
Another pure-logic option: use a 74HC04 and wire a few gates in parallel. Use same supply as MCU (preferably 5V; can also use a separate 5V supply, with 3.3V MCU, with HCT type), and a modest gate resistor to the transistor (say 100/N ohms, for N gates in parallel). The full chip can deliver a couple 100mA total, with less DC current consumption and in about the same space as a discrete (BJTs and resistors) solution.
Tim