When a beta site doesn't generate a lot of daily questions, and gets stuck in beta for a long time because of it, if it were to suddenly to start generating more questions, how long would it have to generate a healthy daily count of questions before it would graduate?
By "launched", I assume you mean graduated, because even a beta site is a fully-functional, launched site even if it isn't (or may never be) "graduated".
I wrote a blog post that discussed this issue:
I've been an advocate longer than I recall for removing that meaningless 'beta' label when it no longer makes sense — I cannot even recall any longer where that argument stands — but we're slowly honing in on those elusive "objective guidelines" for when a site will graduate. Unfortunately, we keep backpedaling on the issue, so I don't know what to say beyond that. Graduation-checks still come up periodically, so if everyone thinks the site "looks good" to go, we submit the site for a design. That is currently the process to graduate a site. I hope we can make it better.