Not really. It happened at about the same rate under Stallings, Bryant, Thomas and Wade - i.e., 55 of the 84 years prior to Saban. Alabama has won more NCs without Saban than any other program other than ND; Alabama, without Saban, has won more SEC titles than any other program (21-14 UGA). Of course, Alabama's total is even more impressive in that those 21 were won in 17 fewer seasons.
Without Coaches Saban and Bryant, Alabama has won more NCs (5, I don't count '41, '45 or '66) than ~98% of college football. Only ND, OSU and OU have won more. That's in 42 (Bryant 25, Saban 17) fewer seasons.
Without Saban and Bryant, Alabama has won SEC titles at a greater rate than every other SEC program - 8 in 50 years.
Alabama is a top 5 program (during the ~60 years without them) all-time without Bryant and Saban. They are transcendent with them. No one else is even close.
In the 66 years without Bryant and Saban, Alabama has 3 (Wade, Thomas, Stallings) coaches (6 or more seasons after 1916) who have an 80+ winning %, no other school has more in 108 years, only 2 have as many: ND and OSU (I included Day whose career is not over, thus could end under 80%), also with 3.
Having said all of that, I agree, Saban is the greatest of all time and will be missed, Coach Bryant is right behind. I was alive and cognizant (though not necessarily unbiased and objective, 9 and weeping at losses when Coach Bryant arrived) during both of their careers.
In summary: However things might look, Alabama has done quite well without a transcendent coach like Coaches Saban and Bryant.
.
