Anyway, it doesn’t say that’s it’s written to “God’s chosen.”
It says, “eklektois parepidemois”, which means “to the chosen sojourners/strangers/aliens.”
Or, if you like, “to the sojourner chosens.” Doesn’t really matter.
Both of them are adjective forms being used like nouns, both of them are the same case and number.
Obviously “chosen by God” is implied, but that’s not what it says. Gotta love all this modern insertion stuff.
Either way… we tend to think of Jews as being the only ones involved in the Diaspora (‘scattering’), and James 1:1 does address a letter to specifically Jewish people. But the whole point of Christianity was that the Gentiles whom God made Christian were now also the chosen people. More to the point, thanks to all sorts of persecution as well as evangelization, Christians were also now scattered in a Diaspora, and Peter could correctly address them as such.
So Peter talks a lot about the former ignorance of those Christians who were once Gentiles, and talks to them directly about how they were called out of darkness into His light, those “who were once not a people, but now are the people of God; those who had not received mercy, but now have received mercy.” (1 Peter 2:10)
So clearly, if he’s addressing the Gentile Christians directly in part of this same letter, his salutation to the “elect sojourners” of Asia et al, must have included both Jewish and Gentile Christians.
Then he goes straight on, and points out that all the Christians are in fact “aliens and sojourners” in their area, the exact same sort of people that God had directed the tribes of Israel to treat kindly, remembering their own nomadic and expatriated days. Because they are aliens and sojourners in Asia et al, they have to be conspicuously non-shady in all their activities, and therefore make the nations who do not know God give glory to God because of their good actions.
(This also could be an implied criticism of anybody who thinks he’s better because he’s a Jewish Christian, and not one of those latecomer Gentile Christians.)