How to drive P mosfet gate to ground in half bridge synchronous DC buck/ boost converter
Didn't do the math but I think that your first transistor is not biased correctly to pull the driver pair.
Why did you put the emitter resistor? You need all the strength available and you are in straight switching mode, there is no need to limit the collector current (more than the collector resistor). 180 ohm into base also seem a little low to me. Did you forget a zero?
Anyway. This kind of discrete driver is only used for huge mosfet/igbt modules (when you have maybe DPAK transistors as gate drivers). It's an interesting exercise but the market is full of dedicate gate drivers ICs (with all the nice protections as a bonus).
Also consider using an N-channel mosfet, they are cheaper and more efficient (from the schematic it seems some kind of buck converter or bridge leg). They are difficult to drive because you need to bootstrap the gate but it's done in the driver IC.