[whatwg/fetch] Remove promise dependency from "Fully read a body" (Issue #1568)

[Fully read a body](https://fetch.spec.whatwg.org/#body-fully-read) is used in contexts that are not necessarily JS related, e.g. loading `<link>`s.
The implementation should go via the streams API without requiring promises internally.
Promises should only be used in JS-related contexts.

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