Migrate injected client to package:web

[annagrin]

The currently used JS interop is restrictive and does not allow for passing structured data between the client and dwds.

Switch to the new package:web to allow for new features.

[bkonyi]

It looks like most of the work for this is done (see #2306) but we need to investigate if there's going to be any breakages in google3.

[bkonyi]

We'll also want to follow up and migrate webdev as well (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.