I'm wrong?
So Ovechkin scored 32 goals and 38 goals. The 2 lowest of his career the two years prior to Oates.
Then when Oates became coach, he scores 32 in 48 games (Rocket / Hart winner) and 51 goals (Rocket Winner).
What's the story then?
Oates did help revive Ovechkin's PP goal scoring. He created what's now known as the "Ovi spot". Before Oates arrived, Ovechkin scored 7 and 13 PP goals in the previous 2 seasons. After he arrived, his PP goals sky rocketed into the 20's. So yes, Oates deserves a lot of credit for making Ovechkin as prolific of a PP goal scorer as he is today.
As far as ES goal scoring goes, he didn't really do much to help him out there. Before Oates, Ovechkin was scoring ~25 ES goals in the previous 2 seasons. He was on pace for the same amount in the lockout season and scored the same amount in the one full season he played under Oates. Oates also forced him to play RW which was obviously not a position he was comfortable with but he did what Oates wanted regardless. Most of his goals came from the left side despite playing RW anyways but Oates was too stubborn to move him back. He was far less dynamic then he could have been if he was allowed to play his normal position.
Oates also had him playing along side of scrubs like Beagle at the time for long stretches. Trotz also played him with Beagle but that was usually with Backstrom centering the line, not Jay Beagle. I remember a stretch in 2014 I think where he didn't score a single ES goal for a month but Beagle continued to center his line despite his slump.