The above circuit utilizes a bootstrap design similar to this next illustration,
Image source: Toshiba - MOSFET Gate Driver Circuit - Application Note
I understand quite well how the capacitor C2 is charged when the low-side MOSFET provides a path to ground. D1 clips this voltage to a safe Vgs for the logic level MOSFETS. Once M2 is switched out it provides a threshold voltage across R2.
Sure enough if I probe Vboot-Vout I can see the clipped bootstrap voltage as it charges to 4.7V.
Now, if I remove the charge path R13 and D4 as well as the zener diode which is no longer necessary.
Now if I probe the same Vboot-Vout I see that the capacitor is charged in reverse to a much lower voltage, averaging about -450mV. Yet the circuit continues to work, almost better than before as the shoot-through is self mitigated since unlike the original circuit this one charges much quicker, albeit in reverse.
Now if I'm analyzing this correctly, I would suggest that the same current that charged C2 forward can be used in reverse when placed across R2, but I can't reason why.