On the contrary, people not being approved has nothing to do with Admins being busy. People are approved daily, but their accounts aren't flagged for review until they meet the roughly outlined criteria that we have posted up for them. In this situation the individual was flagged for review on Saturday night (late, when no one was awake), and marked for approval Sunday. The trophies run on a cron that only runs every few hours (otherwise it would bog the system down constantly rechecking if anyone qualifies for new trophies yet), so the result isn't immediately evident. In this case the user was marked as approved, and then abandoned the place that very same day (within an hour of the trophy notification going through). This wasn't a case of a user posting all over the place and going unnoticed. Excluding their departure post that was made after they were approved, they had made eight posts (which is well within our usual review amount and not an unreasonable length of time or quantity. Accounts are typically reviewed for approval within 5-10 posts of arrival assuming all the outlined criteria has been met). They also sent a PM to myself in which the requirements for approval were outlined to them in the event they hadn't found any of the approval information threads already. This user was not handled any less efficiently than any other user before them.
It is unfortunate they have decided to leave, but we have this information posted everywhere, and an explanation of why we do things this way. We value patience and genuine interest in joining the community (it has nothing to do with assuming people are trolls or trouble makers as the individual in question unfortunately believed, and everything to do with making sure people share our personal values). It's an introduction period for people to take the time to understand the community and to determine if it's a good match for them. I explain this because telling people that they weren't approved because the 'admins are busy' undermines the system and circulates false information. As said above. Approvals are done daily. If someone isn't approved, they simply aren't interacting very much.
That all said, as this user was already marked as approved before they departed, they're more than welcome to change their mind and remain, or to go ahead and depart if they feel that our community isn't a good match for them.