The Rambam (Shevisas Asor 3:7) writes that one is permitted to place some type of wrap around one’s feet on Yom Kippur because the wrap does not diminish the sensation of being barefoot and feeling the hardness of the floor on one's feet.
שהרי קושי הארץ מגיע לרגליו, ומרגיש שהוא יחף
It seems from the Rambam that when evaluating the prohibition of neilas hasandal, wearing shoes on Yom Kippur, more important than whether something is formally defined as a “shoe” in other halachic contexts (e.g. chalitzah) is whether what one is wearing allows for the discomforting experience of being nearly barefoot. If I recall correctly, I remember hearing that the Rav had a hakpada to avoid sneakers with arches – in his view, the fact that one was wearing a non-leather sneaker was insufficient so long as the foot was properly supported and there was no feeling of discomfort. I don't think it is an exaggeration to say that most sneakers today, even non-leather ones, are perhaps even more comfortable than regular leather dress shoes. (For further discussion see the Aruch haShulchan; we do not necessarily adopt this view of the Rambam l’halacha).
See also this post from January which is much more timely now.