I'm completely confused about what conditions need to be fulfilled to graduate a site such as Puzzling SE. Here are some possible answers:
1. Good stats
According to Area 51 stats, all conditions are 'excellent', except for questions per day (9.4 for us, 10 is healthy), which should get resolved in 2-3 months at most.
2. These essential questions
For one, I guess the experienced users are pretty clear about which tags to use where, and questions with bad tagging get edited. Little confusion about synonyms will always be there. (For example, until some time back, there was confusion on tag clarification for probability and probability-theory at Math SE, which had thousands of questions)
Second, I agree that we may have to do more to promote the site outside, but then, I have no idea how I could help. For example, if I share a link to this site on facebook, barely 3-4 people would bother to sign up (and still may not produce good-quality posts).
As for stuff such as logo and elevator pitch, it shouldn't take us more than 2-3 weeks to decide such things, if the mods (on our site, or main meta) actively request us to look at them.
3. Enough rep on the site
Once we are out of beta, the rep required for privileges increases. For example, a trusted user will require 10k rep, not 4k, so we will have to wait till we have a few users above 10k. This, again will happen in 2-3 months time.
So, ... when do we graduate? Will it be within a year of beta (298 done so far), or will it take more time?
Side question: Who decides to graduate us, the mods on meta or the SE staff?