Here is the prologue to Romeo and Juliet, as read by David Crystal in his reconstruction of Elizabethan pronunciation:
The Globe Theatre’s production of the play in 2004 used this “original pronunciation” in three performances only; the following year the Globe mounted an entire production of Troilus and Cressida in OP. These were the first such attempts since the early fifties. They seem to have been fairly well received, but I see no signs that OP is about to become common practice in the performance of Shakespeare.
It does have a nice sound, at least in Crystal’s rendition, and is quite comprehensible after a little adjustment. Easier for an American to understand than a good many modern-day British regional accents. As Crystal notes, it has hints of many modern accents in it, from Irish to West Country to American to New Zealand, without sounding like any one in particular. That’s appealing: it reaffirms our sense that Shakespeare’s language is the common heritage of English speakers everywhere. That would be one clear advantage it would have over RP as a standard for Shakespeare.
On the other hand, the difficulty in “placing” the accent creates other problems. What do you do with country or lower-class speech in the plays? RP versus West Country or Cockney may not be the best solution (especially for non-British productions), but OP seems to offer no solution at all: the accents are either going to sound all the same or the differences will have no meaning that we can identify.
One possibility would be to subtly shade some of the sounds towards RP or some other recognizably “prestigious” accent for the noble characters, leaving the lower class characters to speak a purer OP. There would of course be no historical warrant for this at all, but at least it would make some sense to modern ears, since many of the sounds in OP are distinctly non-posh.
Making puns like Falstaff’s “reason-raisin” clearer is also an advantage to using OP. Restoring perfect rhyme where it has become approximate in modern pronunciation is too, though perhaps not as much as Crystal believes (and much less relevant to the plays than to the sonnets). We can’t really know if Shakespeare always rhymed perfectly. Crystal assumes that he did, and rhyme is indeed a large part of the basis for the reconstruction. It is conventionally held that the first deliberate use of approximate rhyme in English verse was by Henry Vaughan, but who knows? Even if that’s so, half-rhyme has been accepted in English rhymed verse at least since Yeats. To say categorically that rhymes like love-remove “don’t work” in modern pronunciation is a bit extreme.
Crystal’s claim that Shakespeare in OP is much quicker than in modern pronunciation is interesting. It seems to rest on the frequent contractions and elisions in Elizabethan speech (many more than are reflected in the texts, apparently), as well as many vowel sounds being shorter than in modern English (such as the common “me” for “my”). Still it seems a bit subjective. Rapidity or slowness of speech is surely not a purely phonetic matter; it involves prosody and rhythm, which may be harder to reconstruct.
We also couldn’t object to a reconstruction that makes Shakespeare’s verse scan better, but this advantage carries the same caveats as the case for perfect rhyme. Indeed, even if you think Shakepeare’s rhymes were all perfect, you might well doubt that he always preferred his verse to be metrically regular.
In all, the case for OP in Shakespeare seems a good one, but hardly open-and-shut. Using it seems to have been merely one of a hodge-podge of ideas that went into the Globe’s Troilus and Cressida production (which was in modern dress). To make OP really convincing, some fresh overall philosophy of performance that fits with it would be needed. The Bourgeois Gentilhomme production had that in abundance..