It sounds like you are referring to the thin lens formula, but your interpretation of the geometry has led to some wrong conclusions.
It seems to me that if a given lens geometry has a given focal length, then image sensor placed at that focal length would just have a dot at the center pixel as the entire collimated light entering the lens is focused to a point.
Diagrams will help immensely here. You are describing the standard diagram which explains that the focal length of a lens is the image distance to the lens when the lens is focused at infinity:
It sounds to me that you have interpreted this diagram to mean that all the light coming into the lens is focused to the single point at the center of the image plane (i.e., the center of the sensor/film). That is only partially correct: only the collimated light rays that are parallel to the optical axis of the lens are focused down to the single point at the center of the sensor/film.
The following diagram shows red and green collimated light. The rays through the center of the lens are called the principal rays. Any light ray coming into the lens parallel to a principal ray will be focused to the same spot on the image plane that a principal ray intersects.
Revisiting the next sentence from your question,
So really the image sensor should be positioned before the focal point.
Then you are changing the image distance. Moving the image plane closer to the lens moves the distance that subjects are in focus (the focus distance) farther away from the lens. In the case of the above diagrams, that would have us focusing past infinity, meaning nothing is in focus.
In the image below, we are focusing at an object a finite distance away — specifically, the center of Inky's face, between the eyes. Red rays are focusing between his eyes. If the image plane were moved closer to or farther from the lens, then the center of Inky's face would no longer be in focus.
The green lines in front of and behind the image plane (the "eye planes") intersect the red rays in large circles, not points. This is why out-of-focus areas appear blurry. Rays coming from parts of Inky in front of or behind the focus plane (the center of his face) do not converge at a single point in the image plane.