The Rem"a notes that קדיש יתום should be recited after 'Aleinu (O"Ch 132:[סע׳ ?]), however it appears that most authorities require at least three pesukim to precede ק"י (per a comment Double AA once directed at me). According to Nusach Ashkenaz, 'Aleinu ends with only two pesukim: ה׳ ימלך וכי׳ and והי׳ ה׳ למלך וכי׳, one short of the three pesukim minimum, unless we assume that the Rem"a recited אל תירא וכי׳ or added שמע, as in Sephardi siddurim.
Further, the Gr"a notes that 'Aleinu should conclude with the verse of ה׳ ימלך, both since והי׳ ה׳ does not follow the prior passuk in Mussaf for R"H and because he opposes the addition of unnecessary kaddishim (notes to סדור אזור ﭏיהו ad loc.) and Minhag Ashkenaz note that neither passuk should be recited, as they belong to מנין פסוקים for Zichronot, rather than 'Al Kein (ספר דברי קהלת et al.).
As such, why is the common practice (based in the Rem"a, above) to recite Kaddish after 'Aleinu ?