The other answers provided interesting information but missed some of the questions.
I see these questions:
- "So, is this just an internal filter which gradually gets darker near the edges (presumably reducing "bright line" bokeh")?".
Yes, see below.
- "Or is it something more complicated?
- "How big of a deal is this in practical use (or is it mostly a gimmick)?".
IF you like Bokeh that is better than bad but not as good as excellent OR you prefer false appearing blurry blur (blurry Bokeh) then it's for you.
- "Is this something that is always fundamentally better for natural, smooth bokeh ...".
It is not "natural looking" by Eyeball or expensive Lens' definition. From an artistic point of view it might be the most preferable, but why not have V3 (rotating offset Iris, adds depth) and a rotating Lens (controls depth, sheds raindrops for a bonus) to boot.
- "... or can that be achieved in different ways through other types of designs which wouldn't benefit from also having this?".
For Photography it can be achieved using Bracketed Aperture, see near the bottom of this answer.
- "Could the same thing be achieved with a screw-on filter for any lens, or is it essential that this filter be internal?".
A screw-on (or rear-mounted) Filter would cause vignetting, but otherwise produces the same effect.
Normally a small point of light would have an even surface of illumination and when captured with a perfect Lens, exactly focused, you would record a small evenly illuminated point of light.
Portions of the Image outside the Depth of Field (DOF) are out of focus. Points of light that are unfocused become evenly illuminated translucent colored disks, in a 'perfect' Lens.
In an imperfect Lens (virtually every Lens except the most expensive) the disk is not evenly illuminated, often having a ring at the edge and suffering from non-circular shape near the edges of the Image.
Most simply, "Bad Bokeh" (not necessarily "Artistically Ugly", just "Bad" or resulting most often from inexpensive Lenses) is defined as: Bokeh with a sharp edge or rings, produced by Lenses that are molded, polished, and placed poorly, combined with the design of the Aperture Diaphragm having few Blades (an odd number like 7 is low quality, with best results coming from almost twice as many Blades).
Excellent Bokeh (again, not necessarily "Artistically Perfect", just "Technically Perfect" or resulting most often from the most expensive Lenses) is where the point of light, if outside the Depth of Field, is unfocused into a translucent colored disk of light, evenly lit.
Smooth Transition Focus (STF) or a Variable Apodization Filter applies to the translucent disk a modification that might be artistically desirable, a Gaussian Distributed Blur (faded edges, the complete opposite of "Bad Bokeh" with it's hard edge). The radially graduated Aperture creates radially graduated disks instead of evenly lit disks.
Probably most people prefer STF Bokeh over "Bad Bokeh" and even "Excellent Bokeh" from a cost point of view. Nothing replaces perfect Bokeh where perfect circles simply hang in the dark. STF smooths the transition from sharp focus to out-of-focus areas by rounding the edges of the disk (lowering the brightness at the edge).
Correct Bokeh looks like a colored glass disk hanging in the air, the entire surface is the same color and brightness, and there are no visible edges or imperfections on the surface.
In the words of Zeiss' Dr. Stefan Ballmann: "... The number of Iris Blades and the Chromatic Aberration are not the only factors. Without going overboard on the technical details, it is generally the case that lenses with spherical overcorrection tend to exhibit a harsher transition to out-of-focus areas. Lenses that are undercorrected for spherical aberration generally have a much larger depth of field, which creates a softer transition. Distracting double images – in other words sharp edges within out-of-focus areas – are regarded as “bad” bokeh. They are essentially undesirable side-effects that need to be corrected.".
Once you can get single points of light to render in a pleasing manner (in your opinion) either as perfect disks or Gaussian blurred disks, and not as misshapen or outlined disks, then you can use shallow depth of field to blur the background; separating the Subject from the distraction of the unimportant background, and using a Focus Pull to switch Subjects.
It is much as some people like the 'GoPro Look' (barrel distortion) since it makes it look like you're moving fast - well to some it does, to others it looks like a cheap Lens.
The Minolta Dynam / Maxxum 7 Camera has an STF Mode that takes multiple photos and combines them in-camera.
To create the effect with a suitable Lens and Camera simply 'Aperture Bracket' several Photos and using (for example) Photoshop CS4: File/Automate/Photomerge - Source: http://www.dyxum.com/dforum/stf-mode-in-maxxum-7-recreated-for-dslrs_topic54569.html .
Zeiss - Depth of Field and Bokeh by H. H. Nasse: https://www.zeiss.com/content/dam/Photography/new/pdf/en/cln_archiv/cln35_en_web_special_bokeh.pdf
How does ZEISS define Bokeh – An Interview with Dr. Stefan Ballmann: http://lenspire.zeiss.com/en/ballmann-bokeh/