Combining great content with “proper” headings can definitely help rankings. It is rare that anyone would ever need H4, H5, or H6 unless they are marking up a very long piece of content. In most cases, a single H1 and possibly a handful of H2s is all that is needed. Perhaps H3s for longer articles/posts.
Many sites use headers elements for styling rather than to semantically markup the content. This has been done since the beginning of time. Google is aware of this, and they typically shy away from any “signal” that is noisy (which this one definitely would be). Demoting sites because they are using standard HTML elements in their markup seems ludicrous to me. Like many around the web, that SEM agency might be seeing a correlation between the use of H4-H6 and poor rankings, but we doubt very seriously if using h4, h5, and h6 is actually a “cause” for poor rankings.
H1 headings help indicate the important topics of page to search engines.
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While less important than good meta-titles and descriptions, H1 headings may still help define the topic of page to search engines.
We always prefer to use H1 to H3 as it gives good results. Still, h4 to h6 can be used for third priority keywords for good results.