Plugins like Flash, Silverlight, and NPAPI were common at the time, but anything running in a web browser is still a web application. It feels like you're making a distinction without a difference [1]. Either way, the web can performantly do what Quake Live did back then without plugins today, and it can certainly handle the Flappy Bird and Angry Birds style apps that people are playing on mobile devices today. Just take a look at some of the Unity WebGL and threejs demos.
[1] https://en.wikipedia.org/wiki/Distinction_without_a_differen...