[sync] Introduce builder for constructing controllers

The goal of this class is to take a first step towards a simpler way to
construct controllers by:
1) Unifying how sync controllers are constructed (there are currently
   two different to inject dependencies: via BrowserSyncClient and via
   SyncApiComponentFactoryImpl's constructor.

2) Decouple the construction of controllers from the SyncClient class
   hierarchy.

By adopting a builder, telescoping constructors are avoided, a
self-contained object introduced decoupled from SyncClient, and it opens
up various alternatives looking forward.

The new class is temporarily defined inside the same file to keep
the code changes smaller and easier to review, but will be moved to
a dedicated file in follow-up patches.

Change-Id: I3805277042a9a13cb082dbbaf73966344c6252e9
Bug: 335688372
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5658192
Code-Coverage: findit-for-me@appspot.gserviceaccount.com <findit-for-me@appspot.gserviceaccount.com>
Reviewed-by: Maksim Moskvitin <mmoskvitin@google.com>
Commit-Queue: Mikel Astiz <mastiz@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1328950}