If a plane isn't producing more lift than weight it will fall, just like a helicopter. Planes work by pushing a wing through the air, helicopters by spinning it. In both cases the wing has to push down enough air to keep the aircraft in flight.
I am even more confused now.
Helicopters and planes are both pushing down on air to generate lift. The lift generated has to be equal or greater to the weight for the aircraft to fly.
Thrust can mean many things (at least colloquially). As discussed in the quora you linked, a helicopter will have a defined power to weight ratio that allows it to fly (maintain level flight) in its 'normal' flight envelope. There are a number of things the pilot can do that causes the aircraft to 'push down harder' on the air. One of these is flying close to the ground (the ground effect) which is sort of like pushing against the ground as well as the air, and another is by moving horizontally (usually forwards, like a plane, causing transational lift). Both of these allow the aircraft to maintain height while using less power than if it was hovering, but to do so it is still generating enough lift to counteract gravity.
The propeller doesn't contribute materially to the lifting force on an aircraft, while the rotor of the helicopter provides practically all the lifting force on the helicopter.
Both machines need to (somehow) generate lift equal to their weight to stay airborne. The airplane just does this by moving a static wing through the air, which is a much more efficient way of doing it. Its engine/propeller isn't even immediately required to move the wing through the air; once airborne, the aircraft can fly downwards at an angle without engine power to maintain its speed.
You'd say that a helicopter uses "powered lift" while an aircraft does not.