if you used a "WebGL 1st" engine it would probably work. Unity is "compiling" it's project for WebGl so that's one big process that can easily break.
By "webgl-first" you mean three.js or D3? We use Unity for iOS and Android, and we were looking to use a single tool for everything (if possible, of course). Is there any way to turn a simple Unity project into something that won't bring the browser to its knees?