Work towards migrating devtools_app to package:web

[kenzieschmoll]

Migrates most of devtools_app to use package:web. There are a handful of remaining uses to migrate that are not straightforward or where package:web is missing some getters. I documented those on #6606.

I also cleaned up some of the conditional imports while I was here (renaming files, removing unnecessary stub files, etc.).

[srujzs]

Thanks for doing this migration, and sorry for so much feedback!

[srujzs]

Note that this cast is okay in contrast with the cast above. You're casting between interop types and you know it's that type. I think this isn't a lint in the future with extension types.