I think it's incredibly difficult to judge an attacking mid/winger like Sterling against other similar players unless you watch all of them all the time. The role requires you to try to beat players, make key passes, and score goals - all things with a low probability of success. So every decent player who plays this position is going to have highlight reels full of amazing moves where they nutmeg someone or put a defender on his arse or dribble past three players and score, but they could also have lowlight reels full of crap shots, pathetic give-aways, and rubbish passes that go straight out of play.
If you watch Sterling every week, you'll see all of the above, good and bad. Then depending on your point of view, you can reasonably hold any one of a large range of opinions about how good he is. I think the people who are most negative about him are probably comparing him to players after only seeing the highlights of other players' performances, and assuming that they do that all the time, so Sterling should too. When I see Deulofeu, or Nathan Redmond, or Zaha, or Bolasie (etc.) they all look miles better than Sterling based on the few clips of their best moments. However, I appreciate that I don't see the 4 games before their incredible move/goal, or the 4 games after, in which they might very well do f**k all of any worth. Although a different type of player, I think the same's true of Giroud - he scares the crap out of me and looks like a game-winner every time I see him, but the fans who watch him all the time seem far less impressed, as does Wenger, who doesn't even start him half the time. Same goes for Walcott.
So while I'm often unimpressed by the ease with which Sterling loses the ball when he runs at a defender, or his poor shooting, I also appreciate that sometimes he gets past defenders with ease and there was nothing wrong with his shooting against Arsenal (and on his weaker left foot). He's certainly shown he's capable (as have all the other players mentioned above), it's just a case of doing it more consistently in the future.