Migrate webdev to use the new JS interop libraries

[derekxu16]

Separating this out from https://github.com/dart-lang/webdev/issues/2338 so it can be closed. The following is a quote of a comment on that issue that explains which new JS interop libraries replace which legacy ones: "package:js_interop instead of package:js, dart:js, and dart:js_util and package:web instead of dart:html. See https://dart.dev/interop/js-interop#next-generation-js-interop."

[srujzs]

FYI, the old interop and web libraries have now been deprecated. This repo currently has ignores with a TODO pointing to this bug wherever we still use those libraries.