I'd agree with the 10 year timeframe. Before that browsers were basically aiming for HyperCard. It was only after the combination of the WhatWG and the emergence of the app store as an ecosystem threat that the browser vendors have pushed the browser towards OS replacement.

Sorry, the 90s were a real time, and the ambition to replace the desktop OS was already there, as per the ~1995 Bob Metcalfe quip (popularized by Marc Andreesen) that the Netscape browser would soon reduce Windows to "a poorly debugged set of device drivers".

There was a bit of a lull in progress in the mid-2000s, after the fall of Netscape Inc, until the rise of the mobile app threat you mention.

Bill Gates did see Netscape as a potential threat to Windows. Java with Sun's "the network is the computer" motto as well. So yeah, people in the 90s were dreaming big. And then in the 2000s people were complaining that Microsoft set computing back by a decade.