Maybe try this audio from Rabbi Grunstein (hat-tip Joel Rich) on Aleinu before/after Sefiras HaOmer. I think the same logic applies.
As I recall from it: one school of thought is that Aleinu should be in the davening, somewhere near the end. But you can add other prayers afterwards.
The other school of thought (prominent in Nusach Sefard, I think) is that Aleinu, about our faith, should be the very last message [okay there's kaddish] people get before leaving the sanctuary and going out to face the day in a large and spiritually-dangerous world.
Rabbi Grunstein concluded that if this is your custom that's great; if that's your custom that's great; not to make a huge deal about it (or scream at the chazan if he slips up).
Though I'd wonder with shofar-blowing and L'dovid if we'd want to leave people with that final message ...