Your issue therefore is not with the design but rather with the browser. There is nothing preventing you from using a text-based browser or even writing your own.
I do not know if you are a developer, designer, or something else, but you must realize that minimalism is itself a "design sensibility" as you term it.
I have the skills and experience to retool my browsing experience for my more minimalist aesthetic, but most consumers experience the web as something more like an animated version of a supermarket checkout magazine. Since getting a smartphone a few years back, I've been especially struck by how much bandwidth is consumed on transmitting cruft on things like news stories, where the actual text is only a few kilobytes and relevant photographs take up no more than a few hundred kb; but after you pile on all the trackers, divs, ads, etc. etc. the page ends up being many megabytes and can easily take 30 seconds over a typical 3g connection.
To me that's enormously consumer-hostile to very little real publisher benefit; the more shit I have to download to see a page the less likely I am to consume news from that source and the more likely I am to reinforce my ad blockers etc. (whereas I will make exceptions for site that make restrained use of such things so they can get a little extra ad revenue or tracking data value in return for the service they provide).
Recently XKCD made me think about how minimalism fits into design. For example, entirely text-driven:
http://imgs.xkcd.com/comics/the_pace_of_modern_life.png
... and more visually oriented:
http://imgs.xkcd.com/comics/balloon_internet.png
Obviously there are many ways to communicate ideas, and each in various contexts can be quite effective.