That is a great question. And one that is not considered very often.
With the advent of HTML5 the structural design of the page CAN impact the way the page is viewed by the search engines. Notice the word CAN, as these policies are being implemented over time and not all the policies have been implemented as of this writing. But there are some things that you can do that will allow the search engine crawlers to better interpret your page and with that will come better rankings in your search engine optimizations.
For instance, pages in the HTML structure should only have one (1) <h1> heading and that heading should pertain to the main content of the page. It should include relevant keywords that pertain to the main content of the page.
There are many contradictions on this <h1> standard. I have read articles, published by reliable sources that state, "With the advent of HTML5 the answer to this question has significantly changed in both SEO and technical regards. It's now not only possible to have multiple <h1> level headings per webpage that will make sense to search engines, but in most cases it's actually the recommended course." Based on the testing that I have done, I am not convinced that this is true yet. It may be heading in that direction but I do not think we are there yet.
With the advent of the <header>, <nav>, <section>, <article>, <audio>, <video> and <footer> elements assist the assistive devices for people that use the web with disabilities. By using these tags you are more compliant with these types of devices and as such will not be penalized for being non-compliant. Again, indirectly that assists with your overall SEO.
Properly using the tag structure, in the not to distant future will be critical to the SEO success of the HTML5 websites.
Increasing usability, increases SEO. Being able to correctly identify content from the search engines, can increase SEO. HTML5 tags are very SEO friendly for search engine crawlers and is especially effective in audio and video. The better the search engines understand your content, the better it is for your overall SEO.
One thing to keep in mind is that these standards are constantly being updated. One important fact is that most Internet Browsers do not yet fully support all the features of HTML5. They are getting there, but they are not there yet. To be clear, all the browsers will properly render the page, but some of the more obscure elements of the new standard are not yet fully supported. As we get closer to full compliance, if there is such a thing, they will have more and more impact on the overall site structure and how that structure is represented in the search engines.
And we have not even discussed yet all the new features in the CSS standard that pertain to media elements and responsive design. That could take up a few thousands words just brushing the surface of that discussion.
By the way, in no way am I an expert, or even close to an expert in SEO. But I have studied proper site design and structure for many years and reading through all the specifications, structure and styling will become a much more important part of the SEO consideration as we move forward. It only makes sense if you think about it.
So getting back to the initial question.
Does web design relate to on-page SEO and it is an essential element to help your site succeed in SEO?
Without a doubt yes ..... and it will become more relevant as we move closer and closer to compliance to the standard.
I hope this answers your question and helps you understand the topic.