Those proposals seem very limiting and could cause more harm than good. A site should be a representation of that person/organization’s “brand”. It needs to be consistent and complimentary to itself.
A brand can have more than four colors (and I do want to mention that black and white ARE colors). Most branding information I’ve read say that there can be anywhere from 3-8 in a branding palette, but even that gets a little hairy because you can also get into shades of colors, which expands that palette exponentially.
You also seem to be relying on a bit of an outdated implementation of H1. Best practice USED to be a limit of one H1 per page. But with the advent of the headers, footers, sections and articles elements, there can easily be more than one H1 per page now.
I’ll repeat it because it’s important - consistent and appropriate delivery is the important factor. Which color is chosen doesn’t matter.