To clarify here, from https://zulip.com/help/public-access-option#caveats:

> Web-public streams do not yet support search engine indexing. You can use zulip-archive to create an archive of a Zulip organization that can be indexed by search engines.

https://github.com/zulip/zulip/issues/21881 is the issue to track for the native search indexing feature.