P: Gradient editor needs a few improvements

I agree. There have been a number of other legacy issues which are perceived as features by Mr. Cox. At this point I feel they will never be addressed.

In Photoline both linear and cubic interpolation are available for gradient stops. And both look great.

Krita and Gimp offer 5(!) different gradient interpolation methods - and for each gradient segment the interpolation method is independently controllable: linear, curved, sine, sphere increase, and sphere decrease.

I fail to understand why Photoshop's gradients are of such low quality in this day and age, and the only way to control the interpolation is to keep adding more and more stops. There is no need. Mr. Cox mentioned the code behind the gradient interpolation stems from version 5?

Well, about time for an update then. I submit that this thread has conclusively identified an issue with gradient quality in Photoshop, and it is time for Mr. Cox to admit that this ought to be addressed and improved.

For example, here's an simple interpolation that DOES hit the exact same color stops, but has a continuous derivative

It already has a continuous derivative (assuming you set it to 100% smoothness), it just isn't as smooth as an approximating spline.

No I get that you need to hit the color stops, that makes complete sense. But It would be easy enough to construct a spline that hits all the stops and has a continuous derivative, right?

We use a very standard interpolating cubic spline. We need to interpolate the color stops, not just approximate them.

Yes, it is technically true that a piecewise cubic spline interpolation with discontinuous derivatives can have corners. Is there some reason why you prefer to use an interpolation with discontinuous derivatives? The only reason I can think of is that you're trying to avoid the curve overshooting the edges of the RGB cube and 'clipping'. But the price is that your gradients are not remotely smooth even when the RGB line goes nowhere near the boundaries.

Again, an interpolating cubic spline can have some corners when it hits the control points.
It looks like your "smooth" test isn't hitting the control points and just approximating them instead (which gets a smoother curve, but not the colors you specified).

Like I said, it was a quick test. If you really need further convincing I can make you a version where the color stops are exactly identical, but the point here is that there should not be a discontinuity in the Photoshop gradient no matter what the colors are.

And yes, I *do* have it set to 100% smoothness.

Change Photoshop's smoothness to zero and you get a linear interpolation.

Yes, cubic splines that interpolate (not approximate) can have corners where they intersect the control points. Your "smooth" gradient doesn't even have the same magenta color you selected in Photoshop or Illustrator.