Yeah if you put the tags in the content of the posts they will get indexed and search should find them, keep in mind the body of the posts are searched when you search in "posts" so if you are only searching for titles they wont show up.
Locked Tags
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Feature request for the available tags for users to only be the ones added through the admin panel and not free-text entry by users. As an on/off switch in the tags settings in the ACP.
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This has been available in core for a while now. You can set it on the category page.