Based on a thought I had reading through some answers to this question, and many sentiments I've heard over the years about Gene Roddenberry's vision for a universe of total racial acceptance, cultural diversity, etc, why does McCoy always seem like such a flippin' racist toward Mr Spock?

I get that Kirk had a categorical mistrust of Klingons because a Klingon murdered his son, so—okay—he had an emotionally charged reason behind what he said in STVI which blinded him temporarily. Does McCoy have something in has past like this? Is he just a crusty old Southern fossil? Is racism toward other species accepted with the general idea among humanity that they're only down with tolerance among other Earthlings? What's the explanation for McCoy's attitude

Yes, he is a crusty old Southern fossil.

The official Star Trek database (at StarTrek.com) uses your choice of words, almost verbatim, in describing McCoy's disposition:

His temperament was sometimes argumentative, a cynic's outer crustiness masking deep caring beneath the surface. His "old South" roots led to the old-time physician manner of doctoring, with a Southern accent that was most apparent when under stress....McCoy played his role as psychologist expertly to the hilt — especially for the ship's two senior officers. As such an emotional watchdog he was not afraid to take on his captain, but it was his running battle of wits with Spock which became legendary. Spock showed his true feelings, though, as when inviting McCoy down to Vulcan for his "wedding" and in storing his katra with him before a known suicidal saving of their ship before the Genesis detonation.

(Source)

In other words, McCoy and Spock have an underlying respect for one another, but McCoy displays a kind of biting wit and "crustiness" that mask his respect for Spock.

All in all, while humanity has evolved by McCoy's time to have eliminated most public displays of prejudice, this does not mean that every single individual adheres to this, and certainly not in equal measures.

Even by the 24th Century, humanity still has an underlying strain of prejudice. From "Encounter at Farpoint":

RIKER: But your files, they say you're a —

DATA: Machine? Correct, sir. Does that trouble you?

RIKER: To be honest, yes, a little.

DATA: Understood, sir. Prejudice is very human.

In dramatic terms, McCoy can get away with being so irascible because he is the ship's medical officer. His devotion to the well-being of the crew (including Spock) provides a balance to his crusty exterior. The same balance would not work, for instance, in a science or weapons officer.

I have more problem with McCoy being a doctor than a racist.

ASSERTION 1: The doctor does belong as a main character in a TV show based on starship:

JUSTIFICATION: If NASA sends a mission to Mars, it is likely that every member of that mission would be an engineer because, at 220 million kilometres, fixing things would be literally the difference life and death. The people who will be selected for a Mars mission would already be extremely healthy, otherwise they would have washed out in extremely competitive selection process. We know people with slightly high blood pressure don’t make it to become NASA astronauts. So the biggest medical emergency on a Mars mission would be a headache or splinter. There would not be a need for performing a heart bypass suergey. It is precisely that reason why vast majority of astronauts, those who went to moon or even lower earth orbit, were / are engineers. Going to space literally is an engineering thing.

Now I understand, a starship might explore many inhibited planets with different biospheres, with potential for microbial hazards. But it’s nothing pre-contact scans, planet related databases (if available), transporter buffers, force fields and other engineering tools cannot handle. I think Star Trek grossly exaggerates the number of medical emergencies that might occur on starship and this is all just for keep McCoy’s legacy alive in the franchise.

The one thing that second season of Star Trek Discovery is the only time the entirety of Star Trek has got it right is that the doctor is nobody in Discovery, at least recently. I understand that the McCoy character was added just to annoy Spock and not because a doctor on a starship would be any more important than a barber, but thanks to that mistake, we got stuck with a doctor as a main character in every subsequent series.

Star Trek: Discovery, don’t introduce another doctor as a main character again. I am begging you.