I have issues with these criticisms.
> There's also the issue that while the site may look pretty, it requires downloading 5 megabytes of CSS.
3.4MB, and all but 5KB are web fonts which will be cached across the entire site. And while that's heavy for a normal site, it's probably not all that heavy for site specifically about typography. The extra fonts are included because they're used for interactive examples and comparisons, and downloading fonts is cheaper than downloading images.
Web browsers on limited connections are now smart enough to avoid downloading web fonts (on mobile Firefox, there's just a checkbox in settings you can click to turn them off universally). I honestly can't think of a single case where me checking that box has ever broken a webpage. Opera Mini (which is popular in the parts of the world people are most often referring to when they talk about limited connections) disables web fonts as well, so no harm to those users.
Given that there is a readily available, easy to use, safe, user-controlled solution to your problem, I do not see why forcing everyone to use system fonts helps anyone. If you have a problem with the defaults, go to Firefox/Google and tell them to check the box by default.
> Don't ship web fonts except for icons.
My feeling is that on the web we give control to the user, not the developer. And as mentioned above, you can not trust that your icon fonts will be downloaded, so you should not use fonts for icons. It is almost universally better to use SVG for icons. Your icon font is going to be a wrapper around SVG glyphs anyway, so skip the middleman and ship the glyphs directly -- you can still bundle them into a single SVG if you're worried about multiple network requests.
> At every job I've had, designers who talked about typography created more problems than they solved.
Having read the book, none (or at least very few) of the issues you're talking about apply here. The default fonts are heavy and have high contrast (with the exception of footnotes, fair point). They render well on Linux and Windows. They're large and display well on small screens. The book actually goes into detail about a number of your issues, talking about why bad color and unreadable "brand" fonts are a problem. The biggest valid criticism I see here is that links aren't underlined, which... yeah, there are probably better ways to do that, but underlined links do genuinely interrupt reading flow. I view that criticism as a matter of opinion.