Yes, I believe that the "full site" privileges will apply once we get the site design (which as Alex said could well be months away).
For reference, here is the current privileges list. Compare to the numbers for a full site such as Electrical Engineering Stack Exchange.
There are 29 users today who would have access to "edit questions and answers" (2000 rep).
One of the reasons why we took a while to graduate was that, in January 2016, the Stack Exchange people noticed that we had very few people with enough privileges to do things like editing.
In the meantime, before we get our site design done, we all have time to increase our reputation. I notice that quite a few people have quite healthy rep levels, even though they only joined around 6 months ago. You could probably expect those to double in the next 6 months.
Later, the site will still receive a full custom design ...
The word "later" is pretty open. We may well have a year to bump up privileges before the new levels are activated.