This is probably a rather simplistic question to answer, but I've been confused how we go from "on average, the farther away a galaxy is, the faster it’s moving away from us" to "the expanding Universe".

If every galaxy was observed in its present day state, then that conclusion would seem to be valid. However, when we're looking at a galaxy 5 billion light years from us, we're looking at it as it was 5 billion years ago (including the light shift).

Therefore, it seems at least plausible that expansion is slowing rather than accelerating, since galaxies were moving away faster the further back you go in time.

I'm sure the above can be explained, although it may require a lot more time/effort to refute than it took for me to make the argument, but could somebody with more training in the subject point out the flaw in my logic or explain why the expansionist interpretation fits better with observation?

I wonder if this result that there was no singularity true for all theories of inflation or just a subset of them.

Since the very earliest days of Big Bang cosmology there has been a debate over the initial singularity, because nobody likes singularities.

There was a debate about black holes for the same reason. The original Thorne-Hawking bet (scroll down past the more recent one on black hole information: https://en.wikipedia.org/wiki/Thorne%E2%80%93Hawking%E2%80%9...) was driven in part by discomfort with singularities. There was and is a feeling that something ought to make them go away, but we remain unsure if they really will, and until we have a genuine theory of quantum gravity our accounts of what happens at very high densities is going to be a bit hand-wavy.

So this article's suggestion that the current situation is particularly new or different isn't all that accurate. There have always been two senses of the meaning of "the Big Bang" and the degree of comfort people have had with singularities has varied over time (lower in the '60's and '70's, higher in the '80's and '90's, lower again today.)

Nor is it accurate to say that inflationary cosmology is completely uncontroversial and so well understood that we have no choice but to discard the initial singularity. Inflationary cosmologies are popular for good reasons, but they are not the only game in town--we recently saw a thing on Bohmian mechanics go by on HN that suggested an alternative to inflation, for example (I'm far from sold on the work, but use it simply as a way of pointing out that alternatives to inflation exist and can't quite be discarded completely just yet.)