I'd be really interested to see some examples of players who were thoroughly average and not especially efficient through their first 5-6 years in the league who suddenly turned into really valuable contributors.
Stephen Jackson comes to mind as a somewhat similar comp. His points-per-36 floated around 14-16 for most of his first half dozen years (with one spike to 19 in the middle of that stretch), and then jumped up to 18.5+ and stayed there for about 3 seasons. This change correlated with his USG% rate. He's obviously tailed off over the last 3 seasons.
In 'per game' terms, Jackson changed from a guy scoring in the mid-teens his first 6 years (less than that actually in his first couple) to a guy who averaged 20+ for 3 straight seasons.
Efficiency, of course, can be measured in different ways.
Per-minute (and per-possession) production (points, assists) tends to track with USG% rate. So changing roles on a team (or moving to a new team) CAN dramatically change those rates.
Shot efficiency, the efficiency at which you put the ball into the hoop from the field on a given shot is best measured by eFG%. Unless a player makes a dramatic change in his shot distribution - taking more or fewer 3PT shots or getting more or fewer shots at the rim, this doesn't tend to change a lot for most players. It is affected by USG% (because first-option players tend to get better shot quality out of plays), but not as dramatically as 'per-minute' rates.
Jackson's shooting efficiency tended to stay fairly close to his career average of 47% through his 12 year career. That's not terribly efficient. It's not bad. But it's not great. It's gone up & down over the years, but not by a lot and not in strong correlation with the above-mentioned changes to his production.
Joe Johnson is another player who fits this profile. His per-36 numbers his first 4 seasons at Boston & Phoenix were in the low-to-mid-teens. After he went to ATL, they jumped up to over 20 and have generally stay up above 19 per 36 ever since until last year, when he had to share shots with D-Will & others and it dropped to under 16. Again, like with Stephens, his shooting efficiency has stayed fairly stable, though he is a more efficient shooter at just over 49% for his career.
Green's eFG% for his career is about 49% which is actually pretty good. For comparison, Paul Pierce' career eFG% is 50%. Green's eFG% has also stayed pretty close to the latter number, with his career average dragged down a bit by his rookie year (44%, the only year he was below 48.8%). Last year was his career high at 51%.
Basically, eFG% tends to be a little more 'intrinsic' to the player while 'per-minute' efficiencies tend to be strongly correlated with USG%, which depends on a lot of external factors such as the nature of the team you are on and the role you play on that team.
Note - neither of those statements are absolutes. But the effect of role and context on a player's production is very real. This notion that a player 'is who he is' after just a few years only holds true if his role and the context he is filling it in is consistent.
Green was in a consistent role for his first 4 years - 4th option scorer playing as an undersized PF. This last year, his role finally changed, but it evolved over the year. He played both PF and SF last year and his USG grew from that of a 4th option to that of a solid 2nd option player.
Now, Green is about to enter a new context - the team has changed dramatically around him AND his role is also changing as it looks like he will be asked to play a lot at SG as well as SF. From a team perspective, there should be little reason to put Green at PF, given our glut of PFs but it's not clear yet whether that will help his individual production or not.
And, given that Rondo's presence will undoubtedly have a profound effect on the entire team, we may not get a clear picture of just how valuable Green will be long run for quite some time, at least not until Rondo is back.