Because it's a web-based book? I've browsed about and haven't seen a clear explanation. (It took a while to figure out what the hell is going on at that site.)
Using the edges of the pages for navigation as the only option would be ridiculous, but it's not; there are conventional navigation links. Think about reading this on a tablet, where a lot of ebook readers have a convention of turning "pages" by touching the left or right side of the screen, and this may make more sense.
Underlining in the typewriter style is a sin against typography. :) I don't like the way Butterick's chosen to do links here, though; the lozenge mark seems kind of twee, like he's choosing to make a stand against both underlining and color changes at the expense of both readability and learned behavior. (Most of us by now associate both underlines and blue inline text, whether in conjunction or separately, with "click this to make something happen.")
I accidentally navigated away from the page I was reading about 8 times (clicking back each time) through stray clicks on the left or right margin where there wasn't anything there.
It may not work for you on the web, but I've been reading his book with my tablets (both Android and iPad) and I'm finding the experience really positive. Its fresh, clean, and after reading a single page of a chapter I am comfortable with the usability of the rest of the book.
Certainly some 'rules' are broken, and it can ruffle feathers. But I don't think any progress is made without feather-rufflage. This book is a really great example of a readable, usable piece of literature, published on the web, and feels like a next-generation book. I like it.
Apparently he considers underlining a sin against typography [http://practicaltypography.com/underlining.html] as one of the "bad typewriter habits" [http://practicaltypography.com/typewriter-habits.html]. That might be why.
Which is funny, because one of his other rules is against abusing ALL CAPS [http://practicaltypography.com/all-caps.html]--which he does for internal links.
The site is also full of other absurd usability choices, like using the blank edges of the page for navigation [http://practicaltypography.com/how-to-use.html].
The author might know good typography for books, but has gone way overboard applying it to his website.