…giant auger. This design would not have exploited Bernoulli's Principle…
Wings and propellers generate lift by exactly the same principle. The auger would work just like the slender rotating wings of a modern helicopter, just less efficiently.
Generally speaking, anything that flies flies because it generates some force that balances gravity. It can generate the force by using buoyancy or accelerating reaction mass.
Buoyancy is what balloons and airships use. It has great advantage that it does not need any energy. It has disadvantage that it needs huge structure, because air has very low density and thus provides only very little buoyancy.
The other option, moving reaction mass, can be again done in two ways: by expelling material carried for the purpose or by accelerating the surrounding air.
Expelling material carried for the purpose is what rockets do. It is the only currently working way to accelerate in space where outside reaction mass is not available, but for aircraft it is extremely impractical, because a lot of reaction mass is needed and lifting it needs more reaction mass and so on.
So what remains is accelerating surrounding air. And this always boils down to moving a slanted surface through it that always accelerates it using the same principle. Linearly moving wing, rotating wing/propeller, turbine or augur are all slanted surfaces moving through air.
The most important properties of air here are viscosity and inertia. Due to viscosity the flow tends to remain attached to the wing and due to inertia it continues downwards after flowing off the downward-slanted trailing edge.
Then you can use the simple argument of action and reaction—the air is accelerated downward, so the wing must apply force to it and therefore the air applies reaction force of the same magnitude and opposite direction to the wing, the lift—or you can use viscosity, inertia, conservation of mass and conservation of energy (that is Bernoulli's principle) to calculate how the pressure field around the wing looks and notice the lower pressure above the wing then under it.
Neither explanation is more correct than the other; laws of nature all hold at the same time and there are multiple ways to calculate most things. However, Bernoulli's principle is not itself sufficient to explain lift. It is only one of several important properties of fluids that together explain it.