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Sitelinks are far worse at this point, because they don't just appear for brand names. example.com can get sitelinks for a "widgeting" search when there are innumerable other sites covering widgeting as well or better.
To get the search box, you need to search for the company/ organization/ or agency name, or the domain name with the tld, not always the way you'd enter a site.
Nope, I can see it for a quite generic term, think "blue widgets". My site comes up first, will see if it increases traffic, will be quite easy to spot, just look for site: in the logs.