What you want already exists in HTMLForm (although not in its OOUI version). Just use a HTMLMultiSelectField and add 'cssclass' => 'mw-chosen' to it. This will magically turn the usual checkboxes into a Chosen selector (which is pretty much the same thing as OOjs UI's CapsuleMultiSelectWidget). It's used in the OpenStackManager extension, for example on Special:NovaServiceGroup (no uses in core right now, though).

No JS:

With JS enabled:

I see one problem with that JS option: you need to know what you are looking for before typing. :)

I think it will actually show the whole list of available options when
you click the field (like HTML <select> dropdown boxes).

If yes, that's cool! :)

Yep, it should show the whole list when clicking into the input and filter it when typing :)

I implemented this in the last patchset of https://gerrit.wikimedia.org/r/#/c/288197/. Let's merge this back into the original task.