I would guess that the big reason is Microsoft is the owner of Typescript. Apple, Google, and Mozilla probably don't want to be tied to the whims of someone else's language specification. It's the same reason Dart isn't a browser language, NaCl doesn't exist anymore, ASM.js development wasn't widely adopted but sowed the seeds for WASM.
Web standards get worked on by committee where those four companies each have some pull, and standards are voted on.
Node's typescript support is by default just stripping all of the type information out of the source text. There is experimental support for some transformations behind a flag.
https://nodejs.org/api/typescript.html#full-typescript-suppo...