re: Why is Auburn the only member of the "big 6" unable to sustain success?

Honestly, a lot of it comes from coaching snafus.

The last coach to remain long enough and good enough to keep it up was Dye. Remember, Alabama had a similar issue after Stallings left. We had three coaches who coached regular seasons; all had at least one 10-win season (Mike DuBose, 1999 & SEC Title; Fran, 2002; Shula, 2005).

Since Auburn's last run of 10-win seasons and Dye, they've had:
Terry Bowden - 1993, 1997
(Not counting Bill Oliver in this)
Tommy Tuberville - 2004, 2006 (Though to be fair, Tuberville also had 9 win seasons in 2000, 2002, 2005, 2007; he came VERY close to giving Auburn a 4-peat of 10 win seasons and 4-peat of top 10 finishes)
Gene Chizic (2010)
Guz Malzhan (2014, and ongoing)

Honestly, one COULD make the argument that Tuberville might've deserved a better shot; his record was pretty dang good (the best since Dye by a large margin) but his last season could also be seen as writing on the wall that he wasn't able to keep pace.

Gus may very well break the trend this year. I'm inclined to think so. The 10 win streak should be fairly easy, with the larger schedules; the top 10 finish may be more difficult, depending on how well the offense clicks and if the defense is really a step up from last year or not.