Browsing Data Model: De-duplicate deletion logic in Fake BDM

The fake browsing data model re-implemented deletion for partitioned
and unpartitioned data. This duplication in code causes duplication in
effort to maintain the deletion logic and could sometimes be missed.

This CL removes the re-implementation and splits the disk logic to its
own handler for the fake model to override and run synchronously without
handling disk removal.

Bug: 1484482, 1271155
Change-Id: I0c2bf7e20d214bce1b6f857e9f09fdfb9b439314
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5098598
Auto-Submit: Mariam Ali <alimariam@google.com>
Commit-Queue: Theodore Olsauskas-Warren <sauski@google.com>
Reviewed-by: Theodore Olsauskas-Warren <sauski@google.com>
Cr-Commit-Position: refs/heads/main@{#1234440}