Precisely the issue at hand. I completely agree. HTML5 allows us to group or separate content explicitally with appropriate new tags: section & article (header, nav, aside & footer of course included also.) It’s really not that hard.
And with those new elements, the purpose of Headings is now further defined, focused to a narrower, more suitable, semantic and a more precise role. Rather than trying to use them to section split, group AND Title content. A heading is just that, a Heading. A “heading” is simply a Title for a piece of text copy or content. It does not denote a section of content, it merely Titles it.
HTML 5 offers us a clearer way to code semantically and tag group and section our content. The very fact you can use the word “section” so frequently just talking about laying out and splitting up content is testament to its (IMO, perfect) naming and purposful existence in HTML 5. It does what it says, and in itself describes it’s own use. To Section.
Furthermore, with the addition of these new elements, the H1-H6 tags now by impact, are also better described and have become more meaningful and suitable to their real roles; to Title.