Workaround in GIMP:
In GIMP making a custom brush is made especially easy: The content of the clipboard is automatically available in the brushes collection. It can also made permanent by applying Edit > Paste as new brush.
I painted a radial gradient black to transparency and copied it to the clipboard. That's my automatic clipboard brush. Here's one click dot and a random stroke drawn with that brush in otherwise empty layer in another image:
In GIMP alpha (=opacity) is as adjustable as RGB color components. By applying Color > Curves > Alpha I can adjust the falloff afterwards in the whole layer or a selection:
The solid mid part is now wider and the falloff to transparency is made steeper. Adjustable width strokes are possible.
In the next image the result is weird. Solid mid part is made transparent:
One can think it's the same if one applies the alpha curve trick to the gradient dot before or after making it brush. But it isn't. Better control can be got by applying the curves to painted strokes. That's because brushes work in a harmful way. Painted strokes are like numerous consequent clicks, the repeating rate is controlled by parameter named "spacing". The paint cumulates and the result is generally denser than a single click if the automatic repeats overlap.
One can increase the spacing to keep the stroke thinner, but the stroke starts easily look like it's made of separate dots. This happens if the spacing is sparser than the remaining solid looking dot width after the alpha curve trick.
NOTE: In low resolution the gradients are coarse, the adjustment is also coarse.