Search engines such as Google pay special attention to your page’s tag and how much it relates to your content - remember SEO rule number one: if you don’t write it, Google can’t find it! <h1> <h2> <h3> they're held in high regard for SE.
I run through www.stuff.co.nz with website.grader.com, the report tells me all the second level headers H2 and third level headers H3 found by search engines. Obviously, some of the titles within H2 H3 are not the related content which we want to provide to readers, some of the headlines we want to provide to readers are not found by search engine. see the example below:

From the examples we can see, the content in red boxes should be stripped from heading tags, the titles in yellow boxes should be wrapped by heading tags.
I did the same task to www.nytimes.com/, they are doing extremely well.