It's a matter of perspective, and believe me, I have posts talking about when thr FA Cup was a really, really special trophy to win.
Playing for it in and of itself isn't going to be for everyone. I don't believe it's a matter of nationality, but rather how long you've been dialled into English football and whether the significance of what it [the cup] used to stand for, and I'll reiterate that if you're on for a treble or more, it's still regarded as it used to be.
A lot of those players now have a sizeable trophy haul plus a CL, motivation for the FA Cup isn't going to be the driver it once was. If that was the CL final we'd just beat them in, there'd be zero laughing and smiling, that's for sure.
The two major objectives at the start of the season as standalone entities are the league and the CL, and if not the CL, then a sort of regard for the EL, which raises exponentially with progression. The domestic cups register, but not on the same scale as the PL or CL (unless you're on for all 3, or even 4) and it's been that way for an age now, not just for City.
And with regards to direct rivalry, that's like when Fergie suddenly stsrted praising Wenger openly, which officially marked Wenger as washed, but in reverse - when we're a genuine threat to them again in the core objectives of a season (as above), it'll not be the same as it is now.