>Why can't native dropdowns have an integrated search?
At least in Firefox, the normal <select> combobox does search by prefix matching the values of <option> tags as you type.
However it's pretty hard -nigh impossible?- to make a good globally generic search widget. Different datasets warrant widely differing matching rules (prefix matching vs. any place matching vs. any word matching; OR search vs AND search; case sensitivity; i18n concerns; non-alphanumeric characters).