Rich snippets should improve search results overall (see Google's explanation, here: https://support.google.com/webmasters/answer/99170?hl=en), and they do give you some control over what gets returned from a Google search, including a site search. As already noted, for site search this will be independent of your domain's general ranking. Likewise, regardless of ranking, snippets can improve the results returned (or, at least give you some control over same), regardless of where the results are presented (e.g., even on page 5 or 50 of search results).
There is a tradeoff. You can better customize your search results if you use a non-Google database search, BUT, google's site search will generally do a better job of handling synonyms, stemming, etc. (On an extreme, if you were using the Solr/Lucene/Blacklight stack with the necessary customization, you would come out way ahead of Google site search--but only after a significant development effort.) You'll have to decide what makes most sense, or perhaps experiment with what makes most sense for your site.
I could have sworn that Google actually provides an editor that will let you highlight pages on your site, set up patterns, etc., removing much of the snippets coding work for you. But can only find this right now: https://support.google.com/webmasters/answer/146750?hl=en