We're both saying a simple to-and-fro motion won't work. You seem to be saying the trajectory and orientation of the lifting surface need to be properly done. I'm saying that it really doesn't matter.
Have you seen flow fields of nonviscous fluids over an airfoil shape? Begin with a two dimensional irrotational flow field around a circle. Conformally transform the circle to an airfoil shape. The field is still irrotational. Without the rotational component, there is no redirection of fluid velocity and no lift. This is the state of affairs for a perfect inviscous fluid.
Now take a viscous fluid in uniform motion, such as an airsteam first beginning to pass over an airfoil, it's rotation is everywhere zero. It's the stickiness of the air against foil surfaces and into the fluid volume that's required to change the angular momentum of the fluid in the region of the foil required to obtain lift. Once this is established, it will persist around the foil (hopefully, or stall results upon vortex shedding).
So it takes some time and friction between fluid and foil to get this process developed. If you move your foil back and forth, to get lift you have to establish first left handed then righthanded vorticity with each stroke.
This will not happen over a couple cord lengths or less, and the reversal of the stroke will first have to act to reverse the previous circulation. There's all this viscous action going on, half the time reversing what's already there.
I'd imagine that half the time, the foil could be being forced downward instead of up when it's out of phase. I dunno. I'm just guessing about this one. It only just occured to me.