No. Not at all.
The Browser is my portal between sites on the internet. The Nav is my portal between pages on a site. I want them to remain consistent to themselves.
The site has to have a uniform experience across all pages. And the browser needs to stay the same as I switch between sites. I need it to house my various tools and interfaces to best allow me to browse the internet as a whole.
But what's that? If the Nav was in the Browser, it would still be consistent across a single site? Sure, but not all sites are created equal, and I need the Nav to be tailored to that site so I can interact with the site most efficiently. The Nav I use for HN would be wholly inappropriate for YouTube, and the Nav I use on Reddit would be overkill for my personal website.