[path_provider] Remove dependency on engine PathUtils (#11467)

Removes the calls to the Flutter engine's `PathUtils`, and replaces them with direct `Context` calls that have the same logic as the expected codepath for those methods so that there will be continuity with previous releases of `path_provider`. This avoids reliance on `PathUtils` not being removed by R8, and also means that engine changes can't unexpectedly change the paths returned by `path_provider`, since we want to make sure any such changes in this package's return values follow semver.

This does not include the fallback logic from
https://github.com/flutter/engine/pull/30367 because unlike the engine callers, `path_provider` doesn't try to assert a non-null return value, and if the paths aren't available (e.g., due to the device being full) we should return null rather than silently return a different directory than was previously returned.

Fixes https://github.com/flutter/flutter/issues/184750

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