I don't think poetry is equivalent to rap. Certainly there are many rappers who rap without accompaniment, but many of the most famous don't. Rap is sold in separate venues, by separate groups. Maybe slam poetry is somewhere in between the two, but if poetry is all the water on Earth, then rap is more like the Pacific Ocean.
As far as poetry becoming more like therapy or personal angst, I often write poetry to explore an issue I am having trouble conceptualizing, and I consider some of my poetry only for my personal consumption. But I write other stuff I intend to have an audience beyond myself. I dislike how the "therapy" allegation is somehow construed as derogatory. Just because someone intends a piece of writing mostly for themselves doesn't automatically make it inferior.
I've talked to plenty of written poets, slam poets, and people who do both, and I don't think they make themselves inaccessible as a whole. Many of them are incredibly accessible, sharing personal issues and yet having broad appeal in terms of their topics. Also, I've been reading Edna St. Vincent Millay, and I find her as applicable now as when she was alive.
I do think that the prestige culture in the West, especially in America, has shifted since the heyday of poetry. It's not so much that poetry has become less relevant, but that the people to whom it may always have been less relevant have more of a voice in society, and the art forms they most appreciate are now gaining more traction. That's not intended as an insult either to the old culture or the new.