Launch site probably shouldn't show up there, since it would take the user out of the Customize flow and distract from the Save Changes button they need to pick. But if it does show up there, it should probably work.
The Launch Site button is unresponsive.
Chrome / Windows 10

My kid was tinkering around with a new site this weekend and noticed this.
But I've also seen quite a few users complaining of being unable to launch the site in the forums. It seemed pretty straightforward to us, so I couldn't figure out why they were having trouble until my kid pointed out the UX issue.
https://wordpress.com/forums/topic-tag/site-launch/
https://wordpress.com/forums/topic-tag/launch-site/
Launch site probably shouldn't show up there, since it would take the user out of the Customize flow and distract from the Save Changes button they need to pick. But if it does show up there, it should probably work.
I agree just hiding it for now would be best. We might want to add it there later but we need to think about how interactions should look like (i.e. save before exit).
FYI @olaolusoga @johnHackworth @joanrho @dubielzyk
D42920-code has a proposed fix for this, hiding the banner in the customizer.
the fix from D42920-code has been deployed.