Are you trying to make the black dot go to each colored dot in a linear or a curved path? I'm trying to figure out why you're using BezierPlugin.
First, a few notes about why your current codepen wasn't working:
It doesn't make sense to "correlate" the progress in there. correlating is for positional values like x, y, z or top, left where the measurements must factor in them all. Like the distance between two points is the diagonal between the x,y and x,y points. So don't add "progress" in that list.
You mixed x, y, and progress in every object in the bezier, thus you were literally telling GSAP to try setting a "progress" value on the element itself (along with x and y). Obviously the DOM element doesn't have a "progress" value, so that won't work
Again, you were telling GSAP to set "x" and "y" properties of a timeline (along with progress). Timelines don't have x or y properties, so that can't work.
The progress value was animating correctly through each of those values, but they're weighted. In other words, think of an array of x,y points, some of which are very close together and some of which are far apart, and you ask BezierPlugin to animate an object along a path through those points - typically you wouldn't want it to split up the time EQUALLY between each point because that'd make it go really slow between the close points, and really fast between the far-apart points. See what I mean? So BeizerPlugin does a bunch of work to smooth all that out for you. Therefore, when you animated through the progress values, the same thing happened which is what made it SEEM linear. It was doing its job It sounds like you were expecting it to split the time equally.
If I understand your goal correctly (and I probably don't), it's definitely a non-trivial task but it's doable. You could plot a path (like an SVG) using the progress value and maps it to the proportional spot on the timeline, smooth it (to prevent jerky changes in speed) and feed that into a CustomEase. It's definitely beyond the scope of the free help we offer in these forums. If you still need help and would like to chat about hiring us on a consulting basis to help knock this out, feel free to PM me. Or perhaps someone else here wants to chime in with an answer (which would be great).
Good luck with the project!