Migrate webdev to use the new JS interop libraries

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](https://github.com/dart-lang/webdev/issues/2338#issuecomment-2302632635) 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."

Start by reviewing the quoted mapping from issue 2338 and the linked next-generation JS interop guide, then locate legacy imports in the webdev repository. Done means webdev uses package:js_interop and package:web in place of the listed legacy libraries and its existing checks pass.

Written by the indexing model from the issue text.