Why should half-bridge driver have resistance in series with VCC?
That resistor in series isn't just a resistor in series. It is combined with that capacitor to form a low-pass filter.

Notice that the resistor has a very low value in their example, just about 2 ohms. When the gate driver turns on the MOSFETs, it is going to suddenly draw a lot of current as it attempts to charge the gates as fast as possible. The resistor forces it to take it from that capacitor and isolates the VCC line from resonating/ringing. You want a larger capacitor so that your VCC takes longer to dip down. Since the VCC's sudden draw of current only happens for a short time, this will reduce the amount that VCC drops. Remember that with an RC filter, that capacitor will take time to recharge after VCC briefly discharges and so if you make your capacitor too large, it will take a longer time to recharge.

Now, what happens if you don't have that resistor? Your driver will first draw from the capacitor, but the sudden inrush of current makes its way backwards along the power rail in the form of a voltage dip. When that dip hits the source (your regulator or another capacitor), it will likely reflect and become higher as it returns back to the driver's VCC pin. This oscillation could persist for some time and may lead to destructive voltages on the part, depending on the speed (frequency) of the draw on VCC and the geometry of the connection to your power rail (most every piece of copper is a a non-negligible inductor at 10's of ns gate speeds). Adding the resistor in series damps this oscillation (with a corner frequency of \$1/(2\pi RC))\$. In short, that resistor should be considered always necessary, though the value may vary and could depend on the capacitor value you choose.

Another issue you might face is with the length of the trace to the gates from your driver. You must do anything you can to minimize the inductive loop between the gate signals, the GND terminal, and the SW terminal. Keep the copper carrying the signals short, fat, and as close to the other signals as possible. If you don't you'll get the same sort of inductive ringing that you could see on your VCC line and that too could destroy your part.

I would recommend probing at two places: Your Vcc and your two gate signals. You might immediately notice the problem. Based on your failure mode I would suspect ringing on one of your gate lines is doing your driver in.

I also would recommend simulating this circuit with parasitics as well if you have an accurate spice model of the RT9624A that includes VCC inrush. Represent the connection from your power rail to VCC with a small inductor (like 1nH or some other calculated value). Put small inductors on your gate lines (like 1nH or some value you calculate). Just see what happens.