That's exactly what you are doing, whether you realise it or not.
Coca Cola explicitly stated the video in question is NOT part of their training. That couldn't be clearer. The Corporate PR department speaks for the company and will not lie about a thing like this, for fear of legal action and damage to the brand.
So the headline item in this fake story of yours is a pack of lies.
The same statement also said: "The training includes access to the LinkedIn Learning platform on a variety of topics, including on diversity, equity and inclusion. The video in question was accessible on the LinkedIn Learning platform but was not part of the company’s curriculum."
So, it was there, made available and accessible by the Coca-Cola Company for it's employees. Therefore, they clearly "implemented" it and made it available, even if it wasn't officially part of their mandatory company curriculum. The statement implies that it was there but apparently wasn't required. That doesn't make the headline false. The only real question here is whether it was required or optional, but I see that as a relatively minor point - not one that justifies your precipitous charge of it being a "pack of lies." That's just way over the top.
Even the statement from Coke implied that the main part of the story was essentially true, even if it contained a minor inaccuracy.
And yes of course it is furthering a culture war. Why do you imagine this headline is so popular in the far right press? It's because it allows people to feel justified in jeering at attempts to redress the racial balance in society. And that in turn allows people who are a little bit racist to feel that that's OK.
I think you're missing the forest through the trees here, which is a common misperception when people try to address historical and current injustices which have existed in America since before this nation even existed. I don't like the term "culture war," as I think it's a misnomer and doesn't seem to accurately reflect what's going on. I'm also not fond of the term "political correctness," either, although what I see is that the various attempts to redress racism and other injustices in society is that it has become corporatized and sanitized. I think it's a product of insular corporate groupthink, which is why it falls flat and becomes vulnerable to jeering and mockery by the right.