> But maybe one day browsers will have native TS support?

This may already be a thing; node has native TS support by just ignoring or stripping types from the code, TS feature that can't be easily stripped (iirc namespaces, enums) are deprecated and discouraged nowadays.

TS is not actually that special in terms of running it. TS types are for type checking which isn't done at runtime, running TS is just running the JS parts of the code as-is.