It's true, but also true that Phil never has the run of success without Johnny.
There's a good argument to be made that Johnny was the greatest talent evaluator in college football history. It's insane how great he was at selecting assistant coaches. Fulmer's staff in the 90's was loaded with coaches hired by Johnny.
You take Peyton Manning out of the equation and Johnny and Phil have pretty similar runs of success. And don't forget, Johnny was 1 win away from competing for a national title in 1989.
Phil benefited a bit from Alabama being in disarray for most of the time he was there. Albeit, he did have to deal with playing Spurrier in his heyday, but Spurrier was gone from 2002 - 2008, and it didn't stop the decline.
I think the truth of the matter is that Johnny was great at some things Phil was bad at, and Phil was great at some things Johnny struggled with. For a magical period in the 90's we got the "best of both worlds"; Johnny's staff managed by Phil with Phil's top-notch recruiting.