Isn't the proper solution to just use the HTML <select> element, instead of one that recreates the visual, recreates the behavior with JavaScript, and recreates the semantics with ARIA?
<select> is terribly limiting. Options can be a single line of plain text, with some platforms letting you control colour and font, and you get only very basic selection by typing, on desktop platforms only in general.
The “Favorite Animal” dropdown shown in the linked page can be done with <select> with no loss, and therefore I’d say probably should be. But you can often do much better by skipping past <select>: