I am really hoping for some discussion about this, so let me see if I can further "prime the pump".
It has occurred to me that a development in language which might be the inverse of the decline of meter in "poetry" is the rise of literacy and of the printing industry. Novels are not really meant to be read aloud, and the skill involved in reading and understanding them is quite different from that involved in hearing and understanding oral recitation of verse.
I am a slow reader; and, therefore, what I read HAS to be rich to be pleasurable. When I think about the populations that I know of where verse is still prized, I find that they tend to be either the illiterate or the non-academic e.g. rappers and farm folk.
Shakespeare's audience was largely illiterate. The people in it may well have had a MUCH more acute ear for the beauty of verse.