We've beaten LSU in the post-season three times during that period. You can discount those for not happening in "NY6 Bowls" if you want, but: (1) the goal posts can always be moved for anything short of a natty; and (2) those "major bowls" are now pretty meaningless themselves outside of the CFP.
It's just a different model. The SEC headline match-ups are generally tougher than ours, while their cupcakes are also squishier than ours. Check out
Sagarin's SOS rankings for last season. Just looking at the top 20, our schedule (#57) was better than Cincinatti (#91), Buffalo (#76), and BYU (#106), but it was significantly weaker than everyone else's (other than Clemson, thanks ACC).
It's cool that we're one of only three programs who have never played an FCS team before, and I hope they keep that going. But I'm not inclined to beat my chest about it.