Earlier quoted context omitted.
WebGL is basically at native speed. The only issue might be if a GL extension you need is not in WebGL yet, but that is moving along quickly. asm.js can run close to native speed, depends on the JS engine of course, but it's not far off. See for example https://github.com/joelgwebber/bench2d
To qualify, close to native speed means about two thirds of native speed in a best case scenario. Fine for games that don't stress the hardware to its limits, but not for top tier.
For games that are CPU bound, two thirds of speed is the current average on a set of benchmarks on Firefox, yes. But it will vary on codebase, and is constantly improving.
Regardless, you say that top tier games won't work. What is UE4 and the higher-end stuff Unity demoed, if not top tier?