That's like saying "OpenGL doesn't run in a web browser"... and then it did, because someone understood that having 99% the same API and shader language support in a web browser as what is used in native applications is extremely high value.
Many of the people in this thread are looking for the analogous solution to WebGL for Vulkan, and the primary argument for not doing it from the WebKit engineers is that Vulkan is a lower-level API than Metal and DirectX 12, which makes it difficult to impossible to simulate efficiently on top of those APIs; but, this is actually only a problem because Apple refuses to implement Vulkan.