tldr; re "the days of double spacing between sentences" being over, I strongly and sincerely hope not. For me, that little extra inter-sentence spacing is, amongst otherwise well-typeset text, a thing of immense beauty that I will fight in my own way, tooth and claw, not to easily lose.
I rather side with Andrew on this, although admittedly, forums like this are not conducive to achieving an ideal answer. For myself, I very much notice text typeset one way and the other. While my response to "singly spaced" inter-sentence gaps isn’t as visceral as when confronted with lining figures in body text (the horror), given the choice I prefer reading non-French style typeset works.
For myself, I see at least three reasons to prefer wide inter-sentence spacing (W) over non-wide than intra-sentence spacing (N). However, before getting to these, can we agree that, unless we're talking about monospaced fonts, TeX/LaTeX typeset "double spaced" inter-sentence gaps are not really two spaces wide. Guesstimating, they median out at around 1.25 normal space widths or so (actual fount depending).
My arguments favouring W over N (as with all matters about typesetting) draws on the psychological. But rather than using that language, let’s frame this question as an optimisation problem in deference to the more numerous mathematician’s hereabouts. If we can agree that, for most documents, "good" typesetting is congruent with "psychological efficiency", we might then set our objective function to be something like "most information read for least effort expended". In which case, we can ask whether W or N is most optimal. The trouble is that the problem’s dynamical (wouldn’t it be nice if it were static). There are at least two dimensions where the state space explodes. First, we read in different ways for different purposes. E.g., I read blogs in entirely different ways to novels and, again differently, to vegemite jar labels and journal articles. Since the last of these is where I want to be most efficient, if I were to choose a "one rule fits all" approach it would be to maximise article crunching. However, second, we learn to read. The skill is not innate. Having plastic brains, unless we each acquired identical skills and reading habits, we all have (slightly or more greatly) finely different ways of imbibing text. Returning to optimisation, the reading brain flows across the page very quickly/efficiently when the rules it has learnt are satisfied, and (micro) staggers when they are not. The upshot is that, even if we were to incorporate the first constraint into an optimisation model (potentially doable), I hardly know where to begin with endogenising the second into the state space. The fact that brains and reading skills are so diverse really is an awful problem. Perhaps we could collapse the optimisation conundrum by collapsing that dimension down to a point. Of course, to do that we’d need to persuade those who don’t side with us to "shift their thinking" to conform with whichever of W or N that our brain prefers. You know, by calling on unreliable sources (wiki anyone?), or on authorities that serve our needs (Bringhurst?). However, digs aside (maybe Andrew’s feistiness is catchy), what ultimately matters, aesthetics notwithstanding although very much wrapped up in my second point, is the psychological efficiency/experience of reading across the page and, with it, recognition that there are different reading needs and that other brains have learned to read and experience reading in different ways to ourselves.
But wait, there’s more. My second argument builds on my first. Reading is a "pyramidal" experience at every level of cognitive processing. Sentences and easily discerning sentence divisions are every bit a part of that. Now I haven’t any idea whether anyone has performed the experiment, but I’m willing to bet a vegemite sandwich that speed readers as well as highly fluent readers when selected equally from populations trained on W and populations trained on N are as a group on average going to have greater efficiency, less staggering, and consequently greater information retention over W than N text. No, I’m not going to dive into the psych databases to find that out. Not even to defend a vegemite sandwich.
My third argument is less prosaic. I have just spent some time sampling the top 10 tier-1 economic journals and all 10 are typeset in the W tradition. Since they’re what I need to read most efficiently, I’ll rest my case on that. I’d certainly be interested in hearing about which style the journals or books from other home disciplines tend to mete out.