The Bootstrap capacitor will charge through the primary inductor?
Yes, but only at first initial power up where the bootstrap capacitor is fully discharged.
The upper diagonal mosfet initially gets zero gate drive voltage because there is no gate drive power available from the bootstrap.
The lower mosfet is directly powered off the main +12v power supply rail, so when it turns on, it pulls down the source of the upper mosfet through the primary of the transformer.
This only happens the very first switching cycle, charging the bootstrap. Second switching cycle and forever after, both lower and upper mosfets turn on and off together simultaneously.
The bootstrap then relies on the inductor flyback voltage to recharge each switching cycle. If it fails to do so, it will revert to the start up condition and restart anyway.
The upper gate drive circuit has an under voltage lockout so it cannot gradually wimp out from power starvation, it either works properly or it shuts down completely and reverts to start up mode.
Basso's solution is more elegant, but it will work fine without it.