It's interesting that my instructors never use the term "irimi", though I know what it means. I get the feeling they want "ayumi" to be understood as a natural extension of walking.

Well, there's a difference between ayumi 歩み and irimi 入身 - to step or walk vs "enter(ing)/entrance (into) body" which connotates something else altogether... in that you don't necessarily have to physically step in to "enter".

Perhaps it's a play on words - a pun (which the Japanese are fond of)... especially considering the whole Oomoto-kyo connection thing and Onisaburo's interest in chinkon kishin "spirit possession".