Staying at Bama would be admitting that Pete Carroll succeeded at the elite level and he didn't/couldn't.
Pete Carroll didn't have the college success that Saban has had, and Pete Carroll lost as a 40+ point favorite to Stanford. Say whatever you want about Pete Carroll at Seattle, but let's not have revisionist history make us think that Carroll belongs in the same conversation with Saban in terms of college coaching resumes.
(and Pete needed the 3rd stint in the NFL to really be successful. He was 6-10 in his first year in the NFL and got summarily fired. He then went 27-21 in 3 seasons in NE, having a worse record each year compared to the previous. In his first two stints as an NFL HC his record was 33-31, not exactly lighting the world on fire. I doubt Saban spends any time concerned with how anyone views him compared to Carroll)