Animate Calligraphy with SVG

From time to time at Stackoverflow, the question pops up whether there is an equivalent to the stroke-dashoffset technique for animating the SVG stroke that works for the fill attribute. But upon closer inspection, what the questions are really trying to ask is something like this:
I have something that is sort of a line, but because it has varying brush widths, in SVG it is defined as the fill of a path.
How can this “brush" be animated?

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