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.
First of all performance is CPU+GPU, and the GPU side is basically at native speed. Many games are mostly GPU bound, so for them they would run at full native speed.
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?