Not really mental gymnastics. Arsenal expect much higher than 8th so they won't have sat there thinking "he's gotten us 5 points more than last year, great". They made the decision to stick with him regardless of a really poor finish for their club.You can do the mental gymnastics on it till the cows come home, but he improved his league points total year on year, and I'm saying that's why their board stuck by him despite the fans reaction. If they had the same fan reaction and also watched the team drop 3 places he'd definitely have been sacked imo. That's the point here. If Ange finishes outside of Europe and doesn't win a cup he will have gone backwards, it's that simple.
If Ange "goes backwards", in terms of tangible results or points totals, that's still not an open and shut case on sacking him in my view.