[@container] Update the overflowing state via a snapshot client
Use the same ScrollSnapshotClient that we use for stuck query updates.
The snapshot class will be renamed in the next CL to reflect this.
Bug: 367764865
Change-Id: I8cae643860039d3e0c0bb357a19d6807083c06b2
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/5937897
Reviewed-by: Anders Hartvoll Ruud andruud@chromium.org
Commit-Queue: Rune Lillesveen futhark@chromium.org
Cr-Commit-Position: refs/heads/main@{#1371369}