The quote your article led off with was not about race or the importance of integration, but about values. MLK was a minister during the 60s and he saw among the white youth the abandonment of traditional Christian values in favor of "drugs, sex, and rock 'n roll". He was not alone; this message was being preached from pulpits across America and across race.

In my response, the word "conservative" refers to religious/moral values not political party affiliation. MLK challenged blacks to help lead whites back to God rather than join the trend toward atheism, hedonism, and secularism that marked white culture. That challenge remains one in black and white churches across America.

https://apnews.com/article/black-americans-attend-church-pray-more-be5b10abc863c0975c11a92a488e67a3

The fight for equal civil rights was an important part of MLK's legacy, the one most Americans can identify with. However, it is important to remember MLK was a Baptist minister and as such felt that salvation was important and could be evidenced by behavior. As such, his concern with integration is that it could lead to blacks turning their backs on God and instead he hoped blacks would lead whites back to God.

The "burning house" analogy resonates today across America with people of all races.

https://news.gallup.com/poll/506960/views-state-moral-values-new-low.aspx