You nailed it. I went and checked because I'm not
very familiar with Federer racket specs but when he switched from the 90 to the 97, he had to change his racket tension as well.
A lot of people think it's the size of the racket that is the cause of Federer's improved backhand but that's not quite it. It's the racket tension. If you look at his old matches you will notice he used to measure his backhand a lot more and then pick his moments before opening up his shoulders in fear the ball would fly on him but now he just opens his shoulders and swats away non-stop because the tension allows him more control. The trade off is he lost some power on his forehand which is why people often say his forehand is not as great as it once was. He used to string around 22-23 kgs (~48 lbs) and now strings around 27 kgs (~60 lbs).
To me, if Federer had changed his tension a long time ago, this rivalry would have been closer so I don't think it is a true matchup problem. He had a weakness in his backhand that Nadal exploited everytime they played and there is no weakness there with tighter tension.