IDB: shard backing stores across threads, part 2

In this step, ContextImpl => BucketContext flows are made async.
This is achieved with manual `PostTask()` (and friends) calls in
order to support toggling the feature flag. In the final state
it will be more ergonomic to use SequenceBound, but switching to
SequenceBound is not possible while the code still needs to access
the raw pointer (in tests that have not yet been updated and in
production code where sharding is not yet enabled).

Bug: 40279485
Fixed: 327520115

Change-Id: I96e47d3ad3942991e3983f5907051f67e39dc65f
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5332898


Reviewed-by: Ayu Ishii <ayui@chromium.org>
Commit-Queue: Evan Stade <estade@chromium.org>
Cr-Commit-Position: refs/heads/main@{#1267339}