Pages 506-517 (LI - LXII) at the end of the link discuss the historical evolution of Kaddish and the relevant points vis a vis the "Minhag Ashkenaz" in the siddur.
In relation to the possukim after Aleinu, on page 517: (And p84 internal siddur numbering, p93 pdf reader numbering, end of weekday Mincha) it says
According to Minhag Ashkenaz, the possukim "kakatuv bitoratecha" and
"vine'emar" are not considered part of Aleinu, and need not be said,
even in a non-Minhag Ashkenaz synagogue. The possukim should be said
however, if one wishes to recite Kaddish after Aleinu.
(Notwithstanding the fact that reciting Kaddish after Aleinu is quite
problematic according to Minhag Ashkenaz).
Page 507 states clearly that the purpose of Kaddish is to separate between parts of tefilla, so saying it after Aleinu during mincha and ma'ariv is unnecessary (as you aren't separating between two parts of tefilah).
In particular, see page 508, where it says Kaddish is only said after reciting portions of Tanach or possukim, that Aleinu wasn't originally said daily (and the possukim afterwards aren't officially part of Aleinu), and that the practice to recite Shir Mizmor La'asaf(psalm 83) daily (which would justify having kaddish afterwards) was instituted after the ga'onic period and therefore isn't "standard."
It also states that the possuk recited MUST BE JUXTAPOSED to the recitation of kaddish, which is why the possukim in the middle of Aleinu don't count to justify a new kaddish. This is also why we commonly recite "Rav Channanyah Ben Akashya omer" at the end of a shiur - in order to enable the legitimate recitation of kaddish
They hold that the additional Kaddish at the end of davening is a recent innovation and, as such, is outside the "Minhag Ashkenaz," potentially falling under the rubric of "bracha levatalah/she'ayno tzrichah."