With airflow over the treadmill going faster than aircraft stall speed, then indeed yes, the aircraft can take off and the treadmill can be going at zero mph too!
Back in my younger years I spent many an hour up in the Maine bush debating in what situations was it better to takeoff upstream or down stream on a river with a float plane. One gets "up on the step" of the floats based on their velocity relative to the water but the plane flies off the water due to its air speed and you can't make decent airspeed without being "on the step". So often we would begin a take off headed upstream but down wind, get on the step and make a turn back around and takeoff upwind.
I wonder if a float plane could take off from an Endless Pool?
What about a float plane in an endless pool on the deck of an aircraft carrier? Just jack up the water speed of aircraft carrier as high as you can directly into a gale force wind to the point that you exceed the stall speed of the aircraft, and you should be good to go. No need to even spin up the props of the aircraft....it will take off and hover right on top of the endless pool for a moment until it decelerates and hits stall speed and then crashes to the deck of the carrier (so you better turn on the props and generate thrust equivalent to the carrier speed relative to the water after that).
Now we got water in the ocean, water in the endless pool, a sea plane, a carrier deck, motion of the carrier in the x axis (we left out y and z and we left out rho-theta-omega). Can someone ride a Dimond on the carrier deck to see what else happens?