Great idea. I'm all for it. And I don't want to repeat other answers saying that. I just want to add one teeny, tiny tweak:
Along with removing the "beta" label, change the site logo and base design in some visible way. Graduated-but-not-yet-designed sites can all share the same different style, and the change doesn't have to be large (maybe just a change of color scheme), but it would help to do something.
Why?
People who've visited before see "oh, this is different -- not like that three-month-old beta I was just looking at". Make it a little more prominent than just removing the word "beta".
In contexts where all we have is the site logo -- the supercollider list, browser tabs, network profile, tweets, etc -- there's something to convey "not a beta". When I'm scanning one of these lists for a site my universe is currently broken into two groups: blue backgrounds with white characters, and everything else. It'd sure be nice if a newly-graduated site could be part of "everything else" (or, more realistically, a third group).
This doesn't have to be complicated -- maybe just change the hue (blue -> green?). Or, now that betas are back to being shorter betas, bring back the "sketchy" theme for them and use the current beta theme for newly-graduated sites? (That might be harder because of base CSS, though. Just an idea.) I trust the design team to come up with something that works and that wouldn't require weeks and weeks of work to implement.