Correlation is not the same as causation. It's also certainly possible that the types of coaches that bring in top recruiting classes are also the types of coaches that are simply good at coaching the game, hence the positive results. Or any number of other variables I don't care enough to discuss or explore.
I think they're two very distinct skills, and not necessarily correlated. I.e. being a good coach doesn't necessarily mean you're a good recruiter, and being a good recruiter doesn't mean you're a good coach. To be able to do both at an elite level [Saban] is rare.
Now, I agree with you that high recruiting classes don't "cause" program success. If you put me in charge of coaching Alabama's roster, I think Alabama would finish the year with a losing record. Because I have zero coaching experience and all the talent in the world wouldn't mask that.
Nor does coaching "cause" program success. If you put Nick Saban in charge of coaching UAB's roster, he's not reaching the CFP. He might have a winning record against G5 competition, but he'd have a losing record against P5 competition.
The two are related. Talent and coaching together win football games. I personally think coaching is more important than talent overall--as is often seen when a new coach takes over a blue-blood program with an elite roster and fails, or when a new coach [like Jeff Brohm] takes over a team with pretty mediocre talent and actually gets them to outperform. But talent matters, and recruiting rankings are an imperfect but valuable measure of talent.
The only thing I would state is while I don't think "ability to coach" and "ability to recruit" are correlated, I would say that recruiting rankings correlate with coaching talent, for one reason: success recruits itself. You can be a mediocre recruiter but if you can win football games, you don't have to "recruit" talent nearly as much as it comes to you. Winning football games makes the job of recruiting MUCH easier, but I would still say there's a lot of skill involved in recruiting on top of merely kids being attracted to program success.