As part of getting the closure compiler to work, we had to add some asserts and change some class visibilities to @public (see https://chromium-review.googlesource.com/c/chromium/src/+/1021000).

Remove these by fixing the closure compiler to have correct type annotations for functions, or keep them where appropriate. Also, fix function visibilities to be as narrowly-scoped as possible, and remove as many casts as possible.