First thing to realize, is that decent Browsers currently implement only WebGL 1.0 (based on OpenGL ES 2.0) and WebGL 2.0 (based on OpenGL ES 3.0). So that Emscripten SDK may implement only features presented by WebGL 2.0 + extensions, and, unfortunately, Compute Shaders are not presented in any way in WebGL (and there are no more plans to add this functionality in future).
By the way, WebGL 2.0 support has been added to Safari 15 (iOS 15) only this (2021) year, so that even OpenGL ES 2.0 will not work on all devices.
I notice that there is an OpenGL ES 3.1 header in the emscripten
The extra <GLES3\gl3*.h> headers are there not because Emscripten implements all of them, but because this is a common way to distribute OpenGL ES 3.x headers that existing applications may rely on to conditionally load / use OpenGL ES 3.x functions at runtime, when they available.
The documentation says that OpenGL ES3 is supported if I specify -s FULL_ES3=1 (which I am doing).
I think that documentation is more or less clear about what FULL_ES3=1 adds:
To enable OpenGL ES 3.0 emulation, specify the emcc option -s FULL_ES3=1 when linking the final executable (.js/.html) of the project.
This adds emulation for mapping memory blocks to client side memory.
There is no word about OpenGL ES 3.1 here, nor a reason to expect a wonder from Emscripten, as I can barely imagine a reasonable hardware-accelerated way to emulate things like Compute Shaders on top of OpenGL ES 3.0. And software emulation would be certainly of no use for most applications.
WebGPU 1.0 that promised to appear mid '2022, is more capable than WebGL 2.0. So that WebGPU developers already see that at one time native WebGL 2.0 implementation in the Browsers could be replaced by a WebAssembly module implementing this legacy API on top of WebGPU - but in some very distant future. The same could also bring OpenGL ES 3.1/3.2 features to Emscripten SDK - at least theoretically, if somebody will be still interested to work on this.
