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

Use the `omaha` API exposed as free function instead of
the old API going through the `OmahaServiceProvider` class.

The class `OmahaServiceProvider` 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/3115990
  - https://crrev.com/i/4080608
 => https://crrev.com/c/3113975
  - https://crrev.com/i/4080609
  - https://crrev.com/c/3113932

Bug: 1201182
Change-Id: I7f0c87074b669e11fbd857b4621a3fafcbdc653f
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/3113975


Reviewed-by: Olivier Robin <olivierrobin@chromium.org>
Commit-Queue: Sylvain Defresne <sdefresne@chromium.org>
Cr-Commit-Position: refs/heads/main@{#915896}