There was a uservoice and Microsoft was going to turn on where Private Teams would show in Search with approval process.
I've seen the approval process pop up or Teams when you share a link in a team to someone that doesn't have access so this probably could be utilize for the Purpose of adding private Teams to search, however I have not heard anything on this since there was some outrage over it :).
One Method you could do is have your Private Teams added as quick links to a SharePoint page or sharepoint list so it can be searched, and have these links clicked on so it goes to the request access tab of Teams where Owners can approve the access for now until something might get released to accomplish this use case.
