[ios] Use the new `overrides` API in //ios/chrome/browser

Use the `overrides` API exposed as free function instead of
the old API going through the `OverridesProvider` class.

The class `OverridesProvider` is not removed yet to prevent
breaking the internal repository when this CL lands. It will be
removed in a followup CL once the override has been deleted.

This is part of a multi-CL change:
  - https://crrev.com/c/3113926
  - https://crrev.com/i/4078248
 => https://crrev.com/c/3114067
  - https://crrev.com/i/4078249
  - https://crrev.com/c/3113927

Bug: 1201182
Change-Id: I072cf19e892a20a054c34825d1578b367134c49f
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3114067


Reviewed-by: Olivier Robin <olivierrobin@chromium.org>
Commit-Queue: Sylvain Defresne <sdefresne@chromium.org>
Cr-Commit-Position: refs/heads/main@{#914749}