Where to begin?
I will stick with Google because, Who knows what Bing is doing? Your question is actually a rather broad one that requires a bit of understanding. So once again, I will get into a mini-tutorial so that you better understand what you are asking and the answer.
When people think of SEO, they think in terms of one page top-to-bottom and in a vaccuum not fully realizing that no page is ranked or understood alone.
Google looks for code blocks that are at least semi-repetitive following the tradition of the print media. This means that blocks of code that represents a header, navigation, sidebar(s), footers, and the like are easily recognized. This is done not only looking at one page, but many. Google, in effect, segments these and identifies the content of each page. In other words, pages are not really read in a linear fashion from top to bottom except to understand the page. The reality is that these semi-repetitive or repetitive blocks are somewhat ignored when determining the value of a page.
Google does consider some things as particularly important. The title tag is indexed exactly the same way that the content is. I also argue that the description meta-tag is also with far less weight if any. And of course, links are indexed separately. But navigation is not with the exception of links are likely identified as navigational links within the link index and treated somewhat differently. Please understand that these are actual separate indexes. Google calls them barrels, I call them stacks as a throwback to my database days. Each stack has a purpose for making fast references; one for content, one for title tags, one for links, and I rather suspect one for description meta-tags (more on that later). Please notice that nothing else raises to this level. Not even header tags. Keep this in mind.
When Google was a research project, it was found that searches against title tags only yielded almost a 2% more accurate result than searches against the content itself. Google realized that performing parallel searches against the title tags and content, ranking the results separately, then blending the results provided a far better SERP result. This exists still to this day. I believe that this technique has been extended only somewhat to include description meta-tags. I say this because the SERP snippet are highlighted with search terms meaning that Google specifically matches search intent against the description meta-tag. This is not to say, however, that the description meta-tag carries specific weight. I do not see that in the results. What is important to understand is that only one on-page HTML element, the title tag, are important in matching search intent over and beyond the content itself.
Understanding this, You can see that some HTML code elements are treated differently but are not necessarily a part of the ranking of a page. As well, you will find it rather difficult to find in the SERP results matches against headers, footers, sidebars, and the like. This is because these are not content and not normally considered important elements from a semantics point of view. And while we are at it, you will not see search matches against header tags in the SERPs much if at all. Matches are almost always if not always made against content itself.
Now onto the content for a minute.
Conceptually, I want you to think of content in three ways: one, the original HTML as a DOM model primarily used to understand the page elements; two, as a text only rendering of the content; and three, from a semantics point of view. Semantics in search includes more than content, but for now, let's stick with content. It is also true that in effect, the DOM model will identify header tags and other important elements within the content. Also understand, that there are not too many important elements.
When you look at content, you will see some consistencies. There is always a beginning. As odd as this sounds, this is important. There are always header tags generally organized in a hierarchical organizational fashion but not always. There are paragraphs between header tags. Each paragraph has a beginning. And that each set of paragraphs has a beginning starting with the beginning of the first paragraph. Why is the beginning of the content, paragraphs, and set of paragraphs important? Because, in linguistics analysis using semantics, these are important markers. As well, at the very least, in the Google index, term placements are also kept from the beginning of the content. It is not clearly said, but I rather suspect (actually fairly sure) this has changed to include terms within content blocks, header tags as content, and so on.
You are right in thinking that from the very beginning of content, there is a scoring of importance lessening from top to bottom. This is true for a paragraph and sets of paragraphs. This does not meant that content on the bottom of the page is less important. In semantics, it is generally understood that writings follow patterns. The first few paragraphs are only slightly more important than the last few paragraphs, for example. The first paragraph following a header is also important, as another example. The order of content in blocks, content headers, and terms within headers and blocks all score by importance and location using all of the starting points that apply. This means that at any point within your content can a block of content be considered important. Just to add a measure of complexity, sentences are also considered content blocks within paragraph content blocks.
The title tag is a summary of the content. We know that. So is generally the h1 tag. Understood. However, the subsequent header tags are regarded nearly the same for the content blocks in between the header tags. These are generally sub-topics. Each topic, sub-topic, and content block is analyzed for topics, industries, level of expertise, language, and so on using a special set of databases and given scores to identify the value of the content. This means that each and every word of content is analyzed, scored, and understood, as well as sentences, paragraphs, headers, and so on and any part of your content can be seen as important at any point within your page.
Why did I tell you all this?
To show you where SEO importance lies. It lies in the title tag, description meta-tag, header tags, and content. The navigation code has almost nothing to do with search matches but does have something to do with ranking other pages.
What value does navigation have then?
I want you to think of how many SEOs recommend inbound back links and how they say they are important. Why is it that a link is important and are there any clues for internal links?
From the very beginning, when Google was a research project, Google understood that links from one page to another was not just a vote. The notion that a pagerank 6 page with two outbound links passes pagerank 3 to each target page was never really quite true from day one. Certainly, in 2002/3, it was clear that this model was certainly not true. At this time, Google understood that trust and authority caps were needed and that any pagerank value would be dependent upon the value of the link itself. This means that any pagerank passed would be dependent upon any trust and authority cap in place and then calculated against the score of the link which would be less than 1 and therefore any high value page could not pass all of it's value to the pages it linked to and that the value passed depended upon the crafting of the link. A lousy linked with a low score of .1 would not pass the same value of a better link with a score of .9.
From the beginning, Google differed from all other search engines at the time recognizing that the value of the link primarily benefits the target page, but not exclusively. As well, by 2002/3, Google understood the importance of not only applying semantics against the link text, but the content block surrounding the link in the source page and the target page as well. By 2003, these technologies had already been applied and were being improved.
Why is this important? Because the best links are natural in-content links where the link text has the ability to carry significant (as much as possible) semantic scoring. A poor link would be something like read more where as a better link would be within content and would be something like where I describe the basics of semantic scoring in an earlier primer.
Navigational links are different however. They are special. Semantic scoring is not really possible with single word links such as About, Contact, and so on. However, they are strong clues to important content. As with any inbound link passing value, the navigational links are the start of the link chain that helps to score the importance of your pages in a hierarchical fashion. This is why it is important that navigation links are made to your most important pages, but not too many or they will not be counted at all.
Summing it up.
Having said all that, Google does expect certain things but has learned to be very flexible. One of the reasons for the nav tag was so that it would be easier to identify the navigation links. It is not theoretically important if the navigation links are at the top or bottom, however, it is far better to follow tradition. Search engines, Google in particular, are rather old-school for a reason.
One last thing.
The argument that the author of the book you are referring to is fairly old and never really true in the first place. I am aware of this theory. Unfortunately, it is not backed-up considering all of the technology Google has used from the very beginning and how much they actually tell us if we listen. His theory flies in the face of all that Google has said and written they had done many years before. It makes no sense and is counter to all of the search technologies that have existed since 1975 (and yes- that is not a typo). Unfortunately, this is more of the SEO quackery we see here and battle everyday.