Imagine a design that lets you pedal your bike like a regular one but when you need speed you stand on the pedals and push the bike down bouncing it with your whole body weight. and this energy can be transmitted to the wheel- So instead od standing up on an uphill and torture every leg by turns you
just start bounciong your whole weight against the bicycle helped by both legs and arms.

This way might be less stressing on legs, thus giving you more "radius" before you get tired.

I have seen people doing "bunny hops" on bicycles. I think special big tired would help with this method of propulsion - tires that converge on the Hoppity Hop.

I saw this on a kids show yesterday, there is a guy who rents weird bicycles and one of them is a flat bike frame where the rear wheel is spoked so it is off center. To power the bike you push off like a scooter and then hop up and down to power the bike. So interesting (+), but baked (-), but certainly not widely known.

Considering that the bike (and you body) still need to go uphill, why should bouncing your whole body up and down be any less hard than more conventional ways of cycling?

I could only see this work if for some reason the drag would get less for higher velocities, so with every stroke you bring the bike up to speed for a short while, thus being more efficient. But the inverse is the case: Because of the square scaling of drag with velocity, it is more efficient to go at a fixed speed than at a velocity oscillating around this speed.