The answer for my question might be inherent in the question itself- that if I want Google to crawl my site for a custom site search, such as through Programmable Search Engine, that it is those same means that results in my website showing up in public Google searches and if that's the case then great I can move on, but I just want to know if it's possible...
The use case is a website hosted on a subdomain which would host old articles, archived resources, that we don't want to compete with the more up-to-date content on the main domain. However we still want users to access this page via the main site and be able to search through this content. Google Programmable Search is an easy way to have this happen and it's very cost effective.
Currently this subdomain is not accessible by Google or IPs outside of our local network, but as soon as I let Google through to index the page in order to get the searching to work, then doesn't that mean it is then findable on public Google searches?