TypeScript wasn't created separate from JavaScript and then chose JavaScript as a backend. TypeScript only exists to perform build-time type checking of JavaScript. There wouldn't be a TypeScript that compiled to something else, because other languages already have their own type systems.
Runtime type-checking isn't part of TypeScript because 1) It isn't part of JavaScript, and TypeScript doesn't add runtime features anymore. 2) It'd be very expensive for simple types, 3) Complex types would be prohibitively expense as you have to both reify the types and perform deep structural checking.
WASM also is natively supported, and with newer extensions like reference types and GC, we're getting closer to the point where a DOM API could be defined. It'll still be a long while, but that's the long-term direction it's heading in. But even then, you would only see a TypeScript-to-WASM compiler[1] because there's already so much TypeScript out there, not because TypeScript is a particularly good language for that environment. A more static language would be a lot better for a WASM target.
[1]: Porfor is already such a compiler for JS and TS, but it does not do runtime type-checking: https://porffor.dev/