>> People generally want to customize the image, and perhaps the scale of
>> that image.
>>
>> But I see no harm in letting people customize the whole string, if he
>> wants to.
>
> I can't imagine how Customization UI could support customization of
> such strings with text properties and images on them.

Fortunately, now we have customizable icons.  So I tried to do this
with a patch like below, but compilation failed with this error
because tab-bar.el is pre-loaded:

  Loading tab-bar (native compiled elisp)...
  Error: void-function (icons--register)
  (require cl-print) while preparing to dump