I have about 30 or so playtesters whitelisted on my game, and inviting new players would be faster if we could grab the link, even if the game is private.
This would be an easy quality of life feature to enable for closed alpha developers.
My suggestion would be to add a warning message informing the user the game is private, but still providing the link to easily send the link to other collaborators on your project, and playtesters alike.