Brutalist architecture mainly involves raw concrete, proudly exhibiting the modern construction technology of its time instead of pretending to be something else, and more generally avoiding pointless ornament and complication.
It makes sense in architecture, but a similar style in web design doesn't: all web design proudly embraces high tech by its nature, even the simplest examples; ornament can only be simulated, without craftsmanship and actual complication (compare a CSS box border, represented by a few number and implicitly existing even when invisible, with a plaster frame around a door); structures are always simple and evident, differing only in taste.
The point the article tries to make (escaping horribly homogeneous industry trends, with the advantage of having some identity) is valid but unrelated to brutalist architecture.
Looking at http://brutalistwebsites.com/ I see an inconsistent roundup of web design that, since it departs from the mainstream style ("mobile-first", short text, rectangles, centering,...) can be automatically considered relatively cool.
In reality, there are many styles represented: actually mainstream, but accidentally good; glitchy and rough and strange, as traditional for designers showing off; revival of the "Swiss" typographic style that went hand in hand with middle-late brutalist architecture and is somewhat contrary to current trends (close spacing, balanced and regular 2D layout, non-hipster fonts); camp or spiteful imitations of antique and/or aberrant ugly web pages; not self conscious, merely simple and functional.
