But the reason that you can fly slower in the first place is because
the flaps are down. :-) If you look at a CL curve for a wing with and
without flaps you will notice that with flaps extended the curve moves
up and to the left. Up and to the left. (Sorry about the JFK movie
bit). So while the AOA with flaps is reduced for the overall wing, the
local lift coefficient in the section with flaps extended is higher.
The common belief is that downwash causes vortices which in turn
increase drag. Actually, it's the wing tip vortices and vortices bound
to the wing surface that deflects the airflow in the vicinity of the
wing downwards, contributing to the creation of downwash and induced
drag.

Dave