From the user's perspective, isn't emcc a C++ compiler that directly compiles to wasm? There are some internal IRs along the way, but you shouldn't notice them, like you don't notice the GIMPLE IR when using gcc?
The emscripten sdk isn't small, that's true, but that's because LLVM+clang are not small, plus the musl libc and libc++ are not small either. But all those pieces are necessary in order to provide emcc which can compile C++ to wasm. So it can't be just a single repo, multiple components are needed here - in fact, regardless of emscripten, more will be needed soon since the LLVM wasm backend will depend on the lld linker.
What we can maybe do to make this unavoidable complexity seem simpler, is to compile all the necessary things into wasm, and have a single repo containing those builds. So it would contain clang compiled to wasm, lld compiled to wasm, etc. Then the user would check out that one repo and have all the tools immediately usable, on any OS. The wasm builds would be a little slower than native ones, but perhaps the ease of use would justify that - what do you think?