I'd agree with your point about ambitions with respect how the club should act/behave. However given your transfer activity activity, you don't really have the team to aim for the stars IMO. The standards can be really high once the ambition off the pitch matches it, rather than our of the blue, or due to this being Arsenal.
I agree that ambition off the pitch must match the ambition on it but for some reason, I still don't want the manager and the players to get into the mindset of simply fighting for fourth. Push for the title until its mathematically impossible. I just want more for Arsenal and wish I could say Kroenke and the board felt the same.
This is weird logic. Your aims have nothing to do with the results.
Bournemouth could go 'right, we're heading for the title this season lads!' and I'm pretty sure they would still finish mid table. Alright, so Arsenal aren't Bournemouth, but the point is that you don't win just by being overly ambitious and setting big targets, just because you aim for the title doesn't mean you're likely going to get top 4. You'll get top 4 if you're good enough to be there.
Emery hasn't had long at the club and took you over when you were badly struggling, so understandably fans are happy to see genuine progress in a short space of time. It's pretty unrealistic to immediately expect huge improvements in a single season, especially when other clubs invested heavily and have been working on their projects for much longer.
Difference between us and Bournemouth is that we have the resources to aim for the title so that is a fairly disingenuous comparison. When I say aim for the title, I'm not talking about wishing and hoping that we somehow win it. I mean doing everything in our power to win it, buying players, improving our current staff, our fitness methods, all of it. City didn't win the title by simply saying they want to win it. They invested money into the club's training ground, facilities, bought players, brought in Guardiola and went for it.
Why can't we do the same
I didn't expect huge improvements, I don't even expect us to win the league. All I'm saying is that if we set our sights high and work towards achieving that lofty goal then its unlikely that we will fall too far below that goal.
No example really comes to mind and I'm not very familiar with how much time a long distance run should take so this will be arbitrary. Lets say you normally finish a 2km run in 8 minutes but now you want to finish in 6 minutes. When you run its likely you will do so at a pace that will lead to you finishing at or around the 6 minute mark and never really improve from there. Now if you say you want to finish in 4 minutes you will train to reach that goal and obviously increase your pace, you might not get down to 4 minutes the next time or the time after that but if you set it, as a goal then you will get there eventually.