Membership of a restored Israel was neither a precondition nor a consequence of this access, and is not mentioned as such.
Yes, both the Jew and the Gentile now has access in one Spirit to the Father. But Paul continues:
So then you are no longer strangers and aliens, but you are fellow citizens with the saints and members of the household of God…. (Eph. 2:19)
Because they have access to the same God in the same manner, they are not xenoi and paroikoi, both terms which presuppose alienation from Israel, but are “citizens-together” with the saints and oikeioi of God.
It still seems to me that if Paul says that they were once far from Christ, citizenship (politeias) of Israel, and the “covenants of promise”, and then says that they have now been “brought near” to these things by the blood of Christ, he must mean that they have been included in what they were previously excluded from—ie., Israel., the covenant people, from whom the promises and the Christ, etc.
It is not just proximity to God from which they were previously excluded, it is the status of being a citizen of Israel and participation in the covenants of promise (plural, so covenants from Abraham to Jeremiah?). I think this is explicitly stated.