[snip]

No, not so.

The larger the aperture (smaller the f number) the shallower the Depth of
Field is, so the faster the rate at which the image goes out of focus with
increasing distance from the plane of sharp focus, either towards or away
from the cvamera.  So faster lenses can give softer backgrounds when
focussed on the same point with the background the same distance behind that
point.

However, at the same aperture, the DoF is the same regardless of what the
lens's maximum aperture is.

Also, and this is the key point, the quality of the bokeh doesn't depend on
the degree of unsharpness of the image or the extent of the DoF.  Bokeh is
about how attractive the unfocused areas look, so two lenses both at, say,
f4, both focussed at, say, six feet, will produce equal degrees of
unsharpness on a background that is, say, 12 feet away, BUT those
backgrounds may not 'look' the same.  The difference is most obvious in how
any out of focus point light sources are imaged, but overall it is a matter
of how 'creamy' or 'cloud-like' the out of focus areas appear.  (The word
"bokeh" comes from the Japanese for clouds.)

So maximum aperture detrmines how 'fast' you can get an image to defocus
behind or in front of your plane of sharp focus, while bokeh determines how
'attractive' (or, looked at another way, how non-distracting from the
subject) those out of focus areas look.

The key determinants of a lens's bokeh appear to be the degree of correction
of spherical abberration, and the shape of the aperture - which is in turn
determined by the number of aperture blades and how straight or curved they
are.

That's all a simplification of a complex subject.  The factors affecting
bokeh can be measured objectively, but how you feel about the result of all
those factors combining in a given picture is, in the end, a wholly
subjective matter.

Peter