If you have a vertex that is controlled by only one bone, even if the weight of this bone on this vertex is 0.01, the influence will be 100%.
Example: I have this square, the bone has a weight of 0, 0.01 and 1 on its 4 vertices:
If I move the bone, the vertices with a weight of 0.01 will move as much as the vertex that has 1, the vertex that has a weight of 0 won't move:
So if you want to vary the weight of a bone on a vertex, you necessarily need a second bone. If I create a second bone that has a weight of 0.01 on the vertex that has already a weight of 0.01 from the first bone, this second bone will share a 50% influence on this vertex.
For your object, you could create a series of bones, for example 5 bones, parent the object to the armature With Automatic Weights, then give your bones number 4, 3 and 2 a Copy Rotation constraint with bone 5 as Target, and a decreasing influence of 0.75, 0.5 and 0.25:
Here is what it gives: