“Or am I mistaken that King was a Leftist?”

No evidence that he was. Go back a few posts and read Richards writing on revisionist history.
Hoover and others wanted to label Dr. King as a Commie so they could slander him and lay the ground work for a media trial. They were scared of the blowback of trying for a legal trial. IIRC, the King family still thinks that James Earl Ray was a patsy and the support effort behind him was US Government. No more evidence for that then there is that Dr, King was a commie.
But one rumor is considered fact and the other a wild eyed conspiracy.
The lone nut gunman is almost impossible, which is why whenever any government provides that for an answer, a rational man has to look hard at the question. Especially when a government commission puts forth the lone nut theory. The harder the ‘elites’ support the lone nut explanation, the less likely it is to be factual.
Ask a cop that is a friend. He will tell you that if there are loose ends in a case, the DA usually won’t prosecute. Look at the OJ case. To many loose ends. If Jack Ruby hadn’t done the dirty, there would have been one ‘ell of a trial, with a pretty good chance of Lee Harvey walking. It would have been fun watching a Physics Professor trying to explain how a bullet turned in mid-air without touching anything, or having the firearms expert explain how the bullet went thru JFK, turned in mid air, went thru the Governor, then was found on the stretcher at the hospital without a scratch on it. That bullet disappeared later. I’m not a big fan of ‘majic’ bullets and doubt that there are 12 people in Texas that are.
After selling that fresh, 3 day old fish to the public, passing Dr. King off as a commie would be seen as a cake walk. One of my regrets about dying soon is that I won’t live long enough to see some of the 20th century cover ups exposed. That and not being able to visit/urinate on Carter’s grave. SOB has to have made a deal with the devil.