I found that none of the two threads really fixing my problem. They suggest making separate helices which are then spliced together, but this is not a continuously variable pitch helix.
I found a workflow that creates this, however! It's pretty tedious and difficult to control with respect to end dimensions and parameters, but it is possible with some fiddling around.
Step 1: Make a spiral tube on the bottom horizontal plane. Include the outer edge 3D geometry of the tube in a separate sketch for later use.
Step 2: On a vertical plane sketch a profile for revolving around the center axis of the spiral. Make a spline or parabola shape or something else that has an varying slope. This is the effect we'll use to gain varying pitch in the end result. Surface revolve the shape.
Step 3: Surface extrude the spiral up to the revolved shape, using the "To Object" end criterion.
Step 4: Surface extrude a cylindrical shape from the bottom plane.
Step 5: Project (closest point) the intersection curve onto the cylinder, or split the cylinder face using an surface Extension from the intersection curve. The resulting curve on the cylinder will be a helix with variable pitch.
See the screencast for details, even though the demo model is not very impressive. And let me know if it is usable to anyone!