No it’s not. The majority of power comes from the engine. It drives the electric motor mechanically, using it as a transmission. It is not just charging the battery.
Sorry, I meant in the low speed, high acceleration regime (maybe easily confounded with takeoff?). There the engine will turn one motor to generate electricity, which will then power the second motor, like a series hybrid.
No, that's just wrong: at no point is (say) a Toyota Hybrid burning gas to generate electricity to immediately turn another motor that actually moves the wheels.
What do you imagine happens if the battery is low and the car wants to start from a stop? The engine can't shift down to first so it would just stall out if you tried to use it to drive the driveshaft directly, and there's no other source of electricity to put into the motor driving the wheels.
It's a planetary geartrain, with electric motors on the outer and inner, and the gas on the middle. If the electric is _completely_ dead, the gas will just spin without driving the wheels at all. There is no first gear, the mechanical advantage and “clutching” all comes from the inner electrical motor driving the sun gear, while the electric boost/drive comes from the electric motor on the drift shaft (which is on the outermost gear of the planetary set).
There's a video linked elsewhere in this thread that explains how the toyota system works, it's worth watching as it clears up several common misconceptions about the system.
I think what they're trying to say is that there are regimes in which the HSD actually shunts energy from one electric motor to the other (i.e. one of them is "braking" and acting like a generator and the other one driving) and a big part (maybe the majority even) of all power goes through that path. This works even if the battery is empty, because the electricity isn't coming from the outside, it's coming from the spinning ICE.