If you look at the URL when searching "this site" you will see two query parameters: "u" and "k"
osssearchresults.aspx?u=<full web URL>&k=<query text>
You can pass these same params to the Search Center's results pages and it will filter results to that site -- specifically the "u" param.
The downside to this approach is that there is no indication on the page that results are being filtered, and there is no way to remove the filter or switch to "Everything" without changing the URL. In fact, the filter is
kept even when selecting, e.g., People.
You'd have to put some work towards perhaps customizing the search box's display template to control when the filter is used again or give the user a way to remove it. That may be significantly less work than e.g., customizing
and maintaining a new search results page on every site collection.