One problem I have is that it's easily possible to tie yourself up in knots about what you are even measuring when you talk about expansion of the universe. The way Relativity works is that you have to ensure that you are measuring something in the right frame of reference in order for it to make sense, and it's not always clear, to me at least, what that frame is. For the purposes of my day-to-day work gravity isn't even a thing, so I can easily get lost and make mistakes in explaining it when I'm going off half-remembered snippets from lectures years ago. In fact this is the second version of a post, the first version of which I checked and it turned out to be (completely) wrong.
Anyway, I think the key point is that the measurement you describe is an exponential growth rate, and it would not be observed (as far as I know) without something special driving it. If something gave the universe a kick at the start and then gave up (as in, say, a Big Bang + inflation) then the growth rate over a time period P would be, say, a distance D increasing by the same distance D, but then over the second time period the distance would increase by D again, ie a linear growth rate rather than constant doubling.
I don't think you can get exponential growth, as I say, without something special "added on" to otherwise normal physics, such as Inflation (in the early Universe) or Dark Energy or something similar (today).