Yes and no. The issue of keywords had been updated. More and more Google is dealing with concepts in order to bring up a wider variety of candidates for the top ten in order to select the best answer for a query. It's not always necessary for keywords to be hammered on the page.
The header tags should be thought of as a way to provide an outline of what the entire web page is about. The analogy of a pyramid is quite appropriate. A reading of The Elements of Style [bartleby.com] by Strunk and White is recommended to obtain an understanding of the concept of ordering thoughts on a page to form a coherent document.
The W3C has a tool [validator.w3.org] that displays the "outline" of a web page. Basically it grabs your headers and creates an outline of it. A poorly created web page will be diffuse while a well thought out web page will create a scannable outline of what the page is about. This is the O part of SEO, optimization, that gets forgotten, which means making it easy for the search engines to understand what the web page is about.