I'm not sure what the function of the Trench Schottky rectifier is in the circuit but I'll leave that for someone to clarify.
It functions as a "Flywheel diode"; it allows the load current to circulate after the transistors turn off and prevents high voltage spikes that coudl damage the transistors.
The design has an obvious flaw - the gate drive resistors are too high to allow the power FETS to switch quickly; the gate capacitance of those devices is 3.7nF each according to the datasheet, so it takes quite a bit of current to change the gate voltage quickly.
With a 300 Ohm drive resistor the gate capacitance will charge and discharge relatively slowly, causing higher power dissipation in the FETs than if a high current driver were used.
High gate resistiors are not too bad for occasional on-off switching, but at high speeds the inefficiency and power losses could gradually cook the transistors.
The 555 is only rated at 200mA, so could drive 40mA to each device. In high power PWM it's not unusual to have drive circuits capable of 1 - 2A per gate with eg. a 4.7 Ohm or even lower gate drive resistor.
As a compromise, a back-to-back emitter follower driver and going down to eg. 33 Ohm per gate should reduce power dissipation quite a bit?
eg. something like the Q1 & Q2 part of this, added to the 555 output, to boost the current to the FET gate resistors: