This is not ambiguous. You can shove once per turn using an Attack Action, any ONE Attack Action. It doesn't matter how many attacks a player has, ONE of them can be a shove.
Well, given that it’s clearly been interpreted in different ways and not just by folk on this forum I think it’s fair to say it’s ambiguous. “Any one” is not synonymous with “only one”, though I agree it could be read that way. But it seems equally valid to read it as meaning that any individual attack can be replaced by a shove, which consumes just the one attack action it replaces, consistent with other attacks in the same turn also being replaced by shoves. Any clarifications I’ve found seem to support the latter interpretation, see eg
this post and
this article. I’ve not yet found anything similar supporting your more restrictive reading, but the only official-ish ruling I’ve found, the tweet I posted above from Jeremy Crawford, is only slightly less ambiguous than the original rule and doesn’t to my mind completely rule out that only one attack can be replaced. Happy to be pointed at any alternative sources that help clear things up more definitively.
