I would quibble with your proposition that the Klan has not had a "lasting, stable presence in the US." To my mind, being reborn, again and again, despite being (rhetorically, at least) a pariah organization for almost the entirety of its existence, makes it pretty damn lasting!

But I will concede your point. On the surface, it doesn't seem like the KKK, in any of its iterations, was necessarily very successful. However, we shouldn't mistake the appearance of the size or longevity of a particular organization the for long-lasting influence of the ideologies and social forces it represents. Let's take the example of the first Klan to see this process in action. I will defer to W.E.B. Du Bois on this one. This is taken from Black Reconstruction.

"The efforts at reform, therefore, at first widely applauded, one by one began to go down before a new philosophy which represented understanding between the planters and poor whites. This again was not an easy thing for the planters to swallow, but it was accompanied by deference to their social status, by eagerness on the part of the poor whites to check the demands of the Negroes by any means, and by willingness to do the dirty work of the revolution that was coming, with its blood and crass cruelties, its bitter words, upheaval and turmoil. This was the birth and being of the Ku Klux Klan." (Black Reconstruction, p. 623).

So, the significance of the first KKK was its role in the creation of a long-lasting political alliance between the defeated slaver-planter oligarchy and poor Southern whites. This alliance was bigger than the KKK itself; it was a new social compact, centered on whiteness as a political identity explicitly opposed to black self-determination. The KKK was a political-terrorist organization created to this end, which was indeed relatively swiftly put down; but what happened to all of those men who formed bonds and friendships through their shared time in the Klan, or admired the work of the Klan from a distance? Many of them went on into politics, or formed new paramilitaries. The Democratic Party in the "Solid South" was littered with former Klansmen and their like. Wade Hampton, governor of South Carolina, had heavy ties to the white supremacist Red Shirts, for example. There was certainly no country-wide deKlanification; note that the leadership of the Klan was predominately from the slaver-planter oligarchy, which retained and rebuilt its political power steadily over the decades after the end of the war. Du Bois says:

"The masters... forestalled the danger of a united Southern labor movement [between poor whites and blacks] by appealing to the fear and hate of white labor and offering them alliance and leisure. They encouraged them to ridicule Negroes and beat them, kill and burn their bodies. The planters even gave the poor whites their daughters in marriage, and raised a new oligarchy on the tottering, depleted foundations of the old oligarchy, a mass of new rulers the more ignorant, intolerant and ruthless because of their inferiority complex. And thus was built a Solid South impervious to reason, justice or fact." (Black Reconstruction, p. 633)

Can't put it any better than that. The Klan is the icon of this historical moment because it represented its most naked expression. And despite its apparent failure it worked remarkably well. You don't need to have a trillion members; you just need to have a thousand members in the right places. If you can put two former or current Klansmen in a group of ten local judges, for example, then they can dramatically influence law in a given municipality. Do that in a hundred courthouses across the South and suddenly white power is in business. It's pertinent to remember in this moment that according to the DOJ, the FBI, and the Pentagon, white supremacist groups have deliberately cultivated a mass presence in local police departments and the military over the course of the last couple decades.