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. Breaking it down into parts might imply a dissection of motion, whereas we are continually taught to think of everything as one motion. Even the three-step pattern I mentioned is meant to be only a component of a continuous walk, which isn't really supposed to end at the end of practice. I know that sounds weird.
Anyway, it's also interesting that you brought up "suri ashi", which a lot of people mistakenly use when beginning training. The technique in udundi is actually "tachuugwa", walking on the balls of the feet with the heels raised, knees straight, but not sliding along the floor as in kendo. Beginners like me tend to overdo it, but advanced practitioners can do it without looking unnatural.