No, you aren't going to win the league or the CL, but you're still prioritizing them. The FA cup and the League cup, two trophies you are capable of winning, you're disregarding.
"At Spurs." You're basing your expectations on past acheivement. Don't look at what the Spurs sides of the last 40/50 years have done, look at this Spurs and ask "what can this side do?" You have a top four team at your disposal, finishing top 4 with his set of players and this manager shouldn't be seen as an acheivement any more, it should be seen as a minimum requirement. If/when your team lose key players and you no longer have a team capable of finishing top four/winning a trophy, then adjust your ambition and expectation accordingly.
A decade ago Bournemouth were facing the very real possibility of dropping out of the league all together. If they get back to back relegations do you think their fans will say, "well, we are Bournemouth." As of right now they have a Premier League squad, and their expectations come from what they have at this time.
You didn't answer my question. If Arsenal hadn't won those FA cups do you think they would be in a better position now, regularly finishing top four? I'll say it again, winning a trophy and finishing top 4 aren't mutually exclusive things.