Jothar, we aren’t talking about whether its a nice thing to do or if there is an actual source for it, which there is not.
There is a story that the Chasam Sofer davened in some shul somewhere and during Aleinu the congregation turened around to say it. After some investigation he found out that many years prior there were not enough sidurim in the shul and so they placed a large sign with the words for aleinu on the back wall. So even after the signs were taken down people kept on turning around – with their siddurim – do say aleinu.
The point is people do things with no rhyme or reason. And sadly these are the types of nonsense things that people fight for while overlooking actual halochos.
The gemorah in pesachim says that if someone is noheg a minhag shtus (if someone has a custom that is complete nonsence) you could/should tell them to stop. and you certainly dont have to follow the minhag hamokom. (which people dont seem to care about these days anyway)