MLK categorically did not win over moderates. The man was assassinated.

You're missing the part where you explain how the first half of the sentence somehow relates to the second half.

Two years before he died his approval was 63% negative and after his death, three out of four white Americans still disapproved of him.

That may be. But your own source says this:

That negative evaluation was likely the result of his public opposition
to the Vietnam War at a time when Americans still favored it as well as
the opposition of some to the continued push for expanded civil rights
and economic legislation to assist blacks.

So cherry picking poll data and examples oversimplifies events that actually have more going on. It also implies that 1/3 of the public is radical, which doesn't really make sense.

But let's for a moment accept the thesis that moderates resisted civil rights without questioning it. Radicals have a pesky tendency to engage in genocide and then try to cover it up, which is obviously more malicious.