In singarounds and such it's pretty common to have the odd monologue, and I do it myself, but it rarely goes beyond that in my experience. I suspect it's another of those things that are going to creep back into our folioe culture, like storytelling.
But it's definitely true that reciting a song can make people listen to the words in a way they don't when it is sung. Sometimes I've known sigers who will recite a verse or two from a song before singing it, and it can be very effective.
It is something which people who aren't comfortable with their singing voice should consider trying out. (And that doesn't always mean people with bad singing voices - there are many people with great voices that they somehow can't use in public.)