Default heading styles should not have equal top and bottom margin. Headings should be closer to the content they label than to the content they are setting their content apart from.
h1, h2, h3 should not have different styles. it's an anti-pattern that leads to broken accessibility
How does that hurt accessibility? Are you saying people use h3 in order to get its style even when they didn't mean h3?
I think the opposite could happen too - if they all have the same style then people might just use h1 everywhere which is probably just as bad. People tend not to use elements that have no obvious use, like <output> which apparently has better accessibility but does absolutely nothing else, so nobody bothers. The whole "semantic web" thing failed partly because of this.