An onsite search box allows searching for pages that are inaccessible to unregistered users (i.e. search engine spiders). When you have an onsite search box, your visitors will get more search results than in comparison to using Google and the other search engines. If you have an onsite search box and see what users are searching for, you will be able to notice new keywords that are of interest to your audience.

Using the "site:" operator in search engines makes onsite search boxes obsolete is false because generally search engines do not index every single page of a site, even if these pages are not disallowed in the robots.txt or are password-protected. A good onsite search box is really valuable but it is a fact that occasionally onsite search boxes use so imprecise algorithms that using the "site:" operator in Google gives much more reliable results.