I think you're confusing organizing an individual page vs your site. H1, H2, etc, are used to structure/outline your page's content. It tells search engines how your page content is organized, and gives them a synopsis of your page. That's different from what your entire website is about.
Your website as a whole, is about your business/product. Individually, the H1,H2 tags don't say anything by themselves. Let me use an example from schema.org
<h1>Avatar</h1>, the H1 tag doesn't tell us anything. Are we talking about the movie, a emoticon, a computer game character? It's just a structural tag. Your schema, on the other hand, tells search engines about you.
Think of it this way. You have a menu that looks like this:
We have several styles of croissants, both filled and non filled
Do you have a sweet tooth?
Pecan cinnamon rolls
Try our Pecan rolls
Walnut cinnamon rolls
Love walnuts? Try these delicacies.
So what can schema tell a search engine about you? It tells search engines you are an organization, that your company name is "Joe's Bakery", provides your company address/phone, and can even identify your company logo, along with numerous other characteristics.
Ultimately, your website will end up with multiple pages. Together, they along with your schema, will tell search engines what your company is about.
In addition to schema, there are some things you can do to tell search engines more about you or your company, than about products and services. You can also add an "About Joe's Bakery" page and a "Contact Us" page. These will help search engines know about your company as a whole.
In the mean time while you build your site, you can use H1,H2 tags to organize your home page, as long as you remember your trying to tell search engines how your content is organized. It's not just about gaming SEO with putting keywords in these tags. You may also want to spend some time looking at schema. If you're using any kind of CMS, you probably already have a plugin that will help you with your schema so you don't have to write it by hand.