I don't know if I understand your point. Are you saying that the 74% unfavorable view of civil rights demonstrations suggests that Americans disfavored demonstrations but nonetheless were strongly supportive of MLK's speech at such a demonstration?
That strikes me as a level of nuance that is frankly unlikely.
Within a year of that speech the 24th Amendment was ratified to the constitution and the Civil Rights Act of 1964 was passed. He won the Noble Prize a little over a year later. I'd say people agreed with the message above all else because he truly appealed to a shared, universal humanity. This couldn't be done, especially in that era, without a large amount of people supporting this. An Amendment - think about that and what it takes! It almost has to be universal for that to happen. People supported these ideas. It is a myth they didn't and the evidence is the product of them.
I don't think these landmark legal events occurred because people demonstrated so much what the man and his supporters were saying. I believe people miss the forest for the trees and think if they just get a group of people together they're somehow right or will get their way. But it's about what you have to say and how you say it that matters. Peacefully organizing is a great vehicle for that but you still need the goods.
The violence that happened in the later 1960's set so much of it back IMO.
Hmm, to your first part: maybe. Adam Serwer (in that same article) argues that exposure to tales of southern violence, after the Civil War, was instrumental in changing northern Republicans' willingness to push civil rights legislation. So, similarly, in the 1960s.
Yet your conclusion is far too final: it's not a "myth" that people didn't support these changes; some people did and some didn't, as with anything. At one point in the end of 1964, a majority of people oppose the protests that led to these changes.
And in fact, the 24th Amendment faced substantial opposition from southern states; I'm not able to find contemporaneous opinion polls (and I'd be interested if you have any), but it's far from the case that it was without controversy!
I strongly disagree with your last line, however—not because violence is acceptable or productive, necessarily, but because your interpretation exculpates reactionaries who regrouped and pushed back against such changes, which I think is a highly relevant lesson for the Trump era:
Race is such a good predictor of a vote for Trump. The simplest explanation for Trump's rise is that he is a counterreaction to the election of the first Black President.
So too with the success of a cynical Southern Strategy following on the heels of the Civil Rights Era.
Why? If you asked the same question today about BLM you would also see a divergence between the two. Almost certainly not to the degree to which a strong majority favor the notion but disapprove of the demonstrations, but there's going to be a difference.
I don't know offhand of any high quality opinion surveys asking about approval of _demonstrations_ vs _BLM_ in general, so I don't know if your hypothesis is born out.
That strikes me as a level of nuance that is frankly unlikely.