feat: save restored snapshot locally (backport #16060)

[mergify]

This is an automatic backport of pull request #16060 done by Mergify.
Cherry-pick of 81ba019 has failed:

On branch mergify/bp/release/v0.47.x/pr-16060
Your branch is up to date with 'origin/release/v0.47.x'.

You are currently cherry-picking commit 81ba019e5.
  (fix conflicts and run "git cherry-pick --continue")
  (use "git cherry-pick --skip" to skip this patch)
  (use "git cherry-pick --abort" to cancel the cherry-pick operation)

Changes to be committed:
	modified:   snapshots/manager_test.go
	modified:   snapshots/store.go

Unmerged paths:
  (use "git add/rm <file>..." as appropriate to mark resolution)
	both modified:   snapshots/manager.go
	deleted by us:   store/CHANGELOG.md

To fix up this pull request, you can check it out locally. See documentation: https://docs.github.com/en/github/collaborating-with-pull-requests/reviewing-changes-in-pull-requests/checking-out-pull-requests-locally

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[julienrbrt]

utACK.