TL;DR: I want full control of cursor and words during completion.

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Let's illustrate with an example; given the set of allowed values:

enable-test
disable-tests
disable-tests-broken

(These are example values, if it were that simple, I'd change the set of values to easily completable test-enable and tests-disable...)

Classical completion, when typing "test<tab>" cannot do anything. At most it could complete it to "able-test" and the cursor position would remain at the same place. By checking $COMP_TYPE -eq 37 || $COMP_TYPE -eq 63 we can display the full set of wanted completions on completion listing/menu-completion, but it does not help with the trivial completion.

Hence, it would be powerful, if a completion function were able to, after outputing "able-test", place the cursor at the start of the string.
And not only that, when the cursor is at the start, bash will retain any character on the right of the cursor, i.e. if we have now "dis<cursor>able-test", pressing tab currently will not do anything making sense.
You can do COMPREPLY=("disable-tests" "disable-tests-all"), but that just will lead to you having "disable-tests<cursor>able-test". Not user-friendly: user needs to strip the able-test on the right away himself then.

With bind -x '"\e":my_function' it is possible to manipulate $READLINE_LINE and $READLINE_POINT, but that would require me to completely emulate completion for all other registered completions. Which would be absolutely non-trivial.

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I'd propose that, if $COMPREPLY is assigned a string instead of an array, the contents of everything previously parsed into $COMP_WORDS will be replaced by the content of $COMPREPLY. Additionally, the cursor will be placed either at the end of $COMPREPLY or, if an \e is present, that \e will be removed and the cursor placed at that position. This only applies if $COMP_TYPE -eq 9.

Alternatively, to highlight the special case, this may require an additional option being specified to complete (like "-o fullstring").

This way, we can have full control like if it were bind -x, but scoped to completions (and the specific command being completed).