Galatians 6 is indeed relevant as it establishes a precedence for Paul using the term "Israel" to denote a set of persons that is not exclusively Jewish in compositon.
Except that transposed usage makes no sense in the context. Usage is how we determine with what sense a word is used. Hence, the argument is not relevant unless one can exegetically argue that this is how Paul uses the term in Romans 9-11. I can argue definitively that it is not the case.
For Paul to affirm that God has not rejected his people does not force us to conclude that He still has a distinctive role for them.
Actually it does, and he goes even further than arguing for the elect remnant at the present time, but also a future salvation that comes at the fullness of the Gentiles. Have you read Romans 11 all the way through and really studied it? The answers are rather explicitly stated.
Shortly after writing this, Paul gives us the olive tree model where he describes how God will incorporate both Jew and Gentile into a single entity - the church. Paul's position is, I suggest, there is indeed a place for Jews in the church; it is not that ethnic Israel retains special status.
Where does he indicate that the Olive Tree is the Church? The Olive Tree is simply the Covenant People of God, built on the Covenant Promises of God. To this, the Jewish people are the natural branches, and we Gentiles are grafted in as wild olive shoots.
I believe that Paul invokes his ethnicity simply to say that Jews have not been written off by God. But that's not equivalent to an affirmation that they retain special status.
That's not all he says though, hence this isn't a proper interpretation.
So I ask, did they stumble in order that they might fall? By no means! Rather through their trespass salvation has come to the Gentiles, so as to make Israel jealous. Now if their trespass means riches for the world, and if their failure means riches for the Gentiles, how much more will their full inclusion mean! Romans 11:11-12 (ESV)
The "they" is Israel, and they did not stumble in order that they might fall, but rather that through their trespass (the rejection of the Messiah) salvation has now come to the Gentiles. He even adds that this was done in order to make Israel jealous! We can see that this is ethnic Israel because it is distinguished against the Gentiles. Or else, how would the salvation of the Gentiles make the Church Jealous? (makes no sense to import the Galatians 6 sense)
If you'd like I can create a thread on Romans 11 and provide a full exposition that demonstrates my position.
