In order to remain aloft, the wing grabs an air molecule and flings
it downwards with great gusto, that is, with a momentum equal to what
the wing had acquired.


Here's a thought experiment:

You have a vertical, hollow cylinder that contains a vaccum. You have
a circular disk in the cylinder that has a diameter equal to that of
the cylinder and provides an airtight seal against the cylinder walls.

You inject some gas into the cylinder below the disk. The disk will
rise until the pressure reduces to equal the weight of the disk. At
this point, the system is in equilibrium. What momentum transfer is
required at this point to maintain the weight of the disk?