Well, I dunno how well forgotten QMS. or BB are... to me, all these folks were playing some version of the blues but ran through the filter of young folks on acid. And they are all about obscure with folks these days.
My understanding is that Tommy Hall claims to have invented the term "psych rock", but as the wiki points out, folks like the Holy Modal Rounders might have already been using it.
According to my buddy from the elevators (who was, like, 72 when I started playing with him), the Dead an JA really (really) sucked and mostly got traction because they could make drugs show up.
He was saying they were literally doing free shows and handing out pot. As for JA, my buddy was claiming that they were pretty bad, to the point of not understanding how to tune their instruments.
I personally really like the dead, though I never saw them in person, and I am pretty sure there is some amount of sour grapes. But still, they were out there in SF off and on during that time, so I'm just reporting what he said.
Given the nature of this music, I feel that the innovation is what drugs folks were taking at the show... so it doesn't seem odd to me that there are strange mixes of pop, blues, country, bluegrass, etc.
I will say that the elevators might also reasonably be termed Texas Garage rock, which some folks consider a specific genera, and which has a lot pop-ier connotations to me. Those same kids were really into the blues. Moving Sidewalks, for instance, was influenced by the elevators but became ZZ Top rather than something more psychedelic, and there were a whole scene of kids a few years younger in Ft. Worth who skipped the psych and played the blues (Bug Henderson, Stevie Ray Vaughn, and Marc Benno, another weirdo in Kerrville I was playing with for a bit).
Joplin also came out of Texas at that time (she came out of Port A), and as far as I could tell a lot of these folks knew each other and had been playing shows with similar groups in the 65 and early 66.
To your question, I think that the dead really took the psych banner and ran with it. Surely someone would if not them, cause it's fun to drop acid and listen to music. They seemed to have a much, much better head for business than any of these other groups, and so they built a much more recognizable and main stream enterprise out of touring and lighting folks up.