Since shoving deer is very popular right now I think it is worth mentioning that Ayyubids Atabegs are unable to do so.
It would give early Atabegs a purpose, since pushing with three units is way easier than one or two and you could skip making spears to do so. Also it just seems to be an oversight in the first place.
Whilst being pretty self explanatory, here is a short clip I made about the issue: https://www.reddit.com/r/aoe4/comments/1kmcka0/atabegs_are_disney_princesses/