Downwind wouldn't work, you'd just be a very fancy leaf moving with the breeze with no control. It's really the differential between the wind and the water (not so much the keel) that allows the sail to exert a force on the vehicle. Once you're going downwind at the speed of the wind with no reference to the water, you might as well be in still air.

Wind speed typically increases with height and wind direction varies, so you could have a sail or kite that’s in faster wind, but I doubt the difference is large enough in practice.

(in theory, I think a kite up in the jet stream and a fairly large sail near ground level would work for some definition of ‘work’)

Two sail planes on a tether should be able to do it. How the pilots are going to get good at it without getting killed is probably a bigger problem than constructing it.

It's the keel that makes it possible to sail upwind in any useful sense. There are functional equivalents possible e.g. a catamaran can have very narrow-but-deep hulls, but it's the the fact that a keel provides very little drag straight ahead and a lot at 90 degrees which allows you to sail close-hauled.

I remain unconvinced it's impossible downwind, given gliders and parasailing work.

Gliders and parasails are trading height. Boats only have 2D of movement (sort of), they don't have anything else to trade

A keel doesn't just resist sideways motion, it is a foil that produces lift.

Indeed