Anders Hejlsberg is in charge of TypeScript and a apt compiler architect with a proven track record(turbo pascal, delphi, .net/C#).
I think JRuby does a similar thing for Ruby and Java Bytecode. TruffleRuby shows that it is possible to be faster, they do not compile to bytecode and use their own compiler.
There are some dynamically typed languages like Python or Dart that allow (or are in the process to allow) type annotations. But all these languages I know about claim that type annotations do not help performance. I guess you could define a fully statically typed subset of TypeScript that allows you to compile it to fast WebAssembly bytecode. But that would probably also mean to change some semantics: see int vs double for numbers in JavaScript.
Your claim was "it probably wouldn't take that much effort to surpass JS performance with a custom TypeScript -> WASM compiler"
I guess you basically have two options:
1) Implement a JS-Engine on top of WASM. Which is a lot of work. But what could you do to get better performance than current JS-Engines? Every optimization you can implement on top of WASM, you can also implement for native JS-Engines. Quite the contrary it is even harder, since you are an abstraction layer higher and can only make use of WASM opcodes.
2) Define a fully-statically typed subset of Dart and compile to WASM. You could sure do that, but don't forget about e.g. the ubiquitous number type in JS/TS. You may have to use double (almost) everywhere if you want to match semantics of JS. How many non-trivial TypeScript programs could such a subset run successfully? I assume not a lot.
I don't see how you could write a TypeScript to WASM compiler without much effort. Care to explain?