I recently switched roles, and among the seven places I interviewed, none of them seemed to see my then-current browser job as a problem, even though they were not related to browsers. (The closest one was a company implementing a HTTP reverse proxy, and I did not work on the browser's HTTP stack.)
People keep talking about a browser in the modern context but the GP specifically said “early days of the internet” (which, in fairness, would mean pre-web. But I think it’s safe to assume they meant “web” not Internet).
In those days, it was actually a much simpler exercise to write a browser than it is today. I even wrote one! And writing a browser absolutely teaches you how HTTP and HTML worked. Plus a lot of backed development was forms data sent to CGI and thus written in languages we wouldn’t even dream of using for web development nowadays, including C++.
So in the early web, writing a browser absolutely was a transferable skill. It might not be now, but in the context defined by the GP, it was.