If you want your site to appear in search results make sure you have not selected Request public search engines to not display my site
You can request how public search engines treat your site.
Note: This option is only available if "Anyone on the web" can view your site.
In Google Sites, open your site.
At the top, next to "Publish," click the Down arrow .
Click Publish settings.
If do you not want your site available in public searches, click Request public search engines to not display my site.
Use Analytics with your site
You must create a UA (Universal Analytics) code. See this page for information.
VERIFICATION: If you have not managed to verify your Site using the verification method recommended by the Google support article use this one instead:https://support.google.com/sites/answer/9068867,This article uses a different method which is easier.
New Google Sites - Add to Google Search Console
See also this update
Once you get your site added/verified in the Search Console, you can Ask Google to Recrawl Your Site using the "Use the URL Inspection tool" option. Do not use the sitemap option since New Sites do not use sitemaps.
Note that Google Search Console has temporarily disabled index requests at the moment there's no way to actually get your new Google Site indexed.
Index requests are now available.