Is a published site on Google?

A published site is public and crawlable. It ships a sitemap and robots.txt and nothing blocks search engines. Google finds a new site on its own schedule, often days to a few weeks, so it is not hidden, and it is not in Google the moment it is published either. A preview link is the same: public to anyone with the link, just not promoted anywhere.

Put it on Google today

The way a local business site actually gets found is a link from the business's Google Business Profile, the map listing. The owner signs in at business.google.com, opens Edit profile, then Contact, then Website, pastes the site link and saves. The listing shows a Website button within a day or so. On a paying site, Business tab then Manage has a "Put it on Google" button with these steps and a message to send the owner. A site with a custom domain should paste the custom domain.