Follow path controlled by translation either on the x or y axis

Right now an object can follow a path, but the distance on the path can only be controlled by using a value between 0 and 100 of the paths length - which means that if I have a path like the one in the picture and want my red sphere to be at 50% of the x-axis I don't have a way to do this since 50% on the path won't align with the x-axis.
I would like to be able to move an object along a path by controlling either the y or x axis (only one possible of course not both) - for example with a controller and a translation constraint.

So if i move my blue controller from left to right, the red sphere would stay on the same x position as the controller and move up and down following the grey path. For scenarios where the sphere can't follow the controller (if for example the path is u shaped and the long sides are parallel to the chosen axis) the sphere should follow the controller as far as possible and then stay at the last point where it can no longer follow - similar to how the distance constraints work in some scenarios.

thx for taking the time looking into this :)