I was recently in Jerusalem for a period of time and while I was there, I would join a minyan at a nearby yeshiva for mincha every day. At the end of mincha, the person davening from the amud would conclude Aleinu by saying
ככתוב בתורתך ה׳ ימלוך לעולם ועד
He would not say the second verse that is included in every siddur I have ever seen:
ונאמר והיה ה׳ למלך על כל הארך ביום ההוא יהיה ה׳ אחד ושמו אחד
Kaddish would begin immediately after he finished (i.e. there was no time for someone to have said the verse quietly before beginning kaddish). The shaliach tzibur was different every day, but they always followed this pattern. The siddur they were using was סדור כלילת יופי - אשכנז which does include the skipped verse. Is this a recognized custom? If so, what is the reason for skipping that part of the text?