[OTP] Manage GMailOTP cache in the backend instead of OneTimeTokenService.

We are preparing to remove OneTimeTokenService as an intermediate layer,
so the cache has to be moved a layer down.

Bug: 556170395
Change-Id: I30802ad3435e24a6aa013a5d2c7167656a6a6964
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/8425408
Commit-Queue: Bekzhan Kassenov <bkassenov@google.com>
Reviewed-by: Marc Treib <treib@chromium.org>
Reviewed-by: Jihad Hanna <jihadghanna@google.com>
Cr-Commit-Position: refs/heads/main@{#1704925}