This replaces our current DocSearch 2 implementation with DocSearch 3, including a new dialog that pops up that can save recent searches in local storage. I've lightly restyled things to better suit our docs, but could be down for more changes. Right now this is all done via CDN links, but we could also bring this in via npm or a manual copy-pasta if we prefer that.
Current known issue w/ Safari: https://github.com/algolia/docsearch/issues/1260.
Base branch is #35736, but that'll change after we merge that.
Fixes #33338 (closed)