In FM, he would. I'm disregarding the fact that he's failed the long term expectation of winning the league within 2 years as Arsenal haven't won for the last 3.
Make it to FA Cup final: No
Make it to league cup final: No
This represents a serious fundemental problem in FM. You can fired for doing very little wrong. Lets say Man U pip Arsenal for the title (quite possible) and Arsenal eventually lose in the Champions League (more than possible), would you really think the Arsenal board would fire Arsene Wenger?
I think, that even if you only satisfy one board expectation, if you're pretty good otherwise and definitely improving, the board have no grounds to fire you. I got fired from my 8.01 Valencia game because I didn't make it to the Spanish Cup Final or the Champions League Finals and lost the Super Cup (all due to Barcelona...). I managed to win the league, but I did not get a new contract. Do you think any manager would EVER get fired after they win the League?
I think FM needs to:
A. Either make board more patient or make there be an unsackable option.
B. Add "focuses" to the game. You can choose to focus on one target (IE, FA Cup, League, CL, etc). and the expectations will be higher for that, but not as high for the non-focuses. It can easily be implemented and work like when you can switch the squad bonuses, budgets around.
C. This would be hard to code, but have the board take HOW you didn't achieve your expectations into consideration. If half my squad gets injured and I lose in the early rounds of the Spanish Cup to Barcelona, should the board be THAT unhappy? Obviously not, but they don't take that into consideration. Also, the first season Barcelona finished with over 100 points... but I managed to win the second season to not get fired (what I said before did not happen until a few seasons later). Real Madrid's manager got fired even though they had 90 or so points. I felt bad for him!
I was really hoping this would be fixed in the patch. I'll give you an example of the playability differences between 07 & 08. In 07, I never had to reload, while in 08, I need to save before every match so I can satisfy board expectations.
Cmon, SI. You know it's a problem!
PS Patch 8.02 gameplay is MUCH better, especially the Super Keeper issue. Good job on that!