Yes, I think so. Not sure why they did it that way. Maybe they couldn't get a good high current path in the ic. The TO-220 only has the push pull output.
One advantage to having a separate output for each MOSFET is that it would be much simpler to provide the driven MOSFET with an asymmetrical drive. simply by using two different value resistors. For a symmetrical drive that still requires a resistor, two resistors could be used instead and that would reduce power supply current shoot through if the internal driver doesnt have any dead time, and that could reduce the bypass capacitor size.
Looking at the data sheet it looks like the two outputs are to be connected together and it wont work very well without using both outputs one way or the other to drive one ore more MOSFET gates.
The shoot through can happen if one transistor stays on (either fully or partially) when the other transistor turns on (or is turning on). The way the schematic is drawn it appears that this can happen, but it could be just an over simplified schematic. If there is any shoot through you'll see the power supply pin dip low for a short time period even when the lower transistor turns on. You might want to check for that. There must be some management though or the device would burn up.