Paul’s heart is for “Israel” (not specifically named in Rom. 10:1-4 but implicit); his prayer is for their “salvation.” Beyond the Exile, if we follow the motion of Paul, there will be Covenant Renewal, and that will occur in Christ.
Israel is, according to Paul and not unlike Paul’s own past, zealous but ignorantly so (Rom 10:2). What Paul believes will lead from that “ignorance” to “knowledge” is not gnostic but instead publicly available: God’s Covenant Renewal is in Christ.

Now Paul says something that long ago was given expression by EP Sanders in this way: The problem with Judaism, for Paul, was that it was not Christianity. Let me unfold this to say what I think Paul is saying in Romans 10:3 (and we’ll leave 10:4 for Monday):
The Exile is ended and the Covenant Renewal is established in Jesus the Messiah. Therefore, adhering to the righteousness through the Torah is a denial of God’s righteousness. There is no other way: God makes things right/ declares people right in and only in the Messiah. Thus Romans 10:3.
I summarize how Wright understands 10:3: Ignorant of God’s righteousness (not righteousness that comes from God but the righteousness that is God’s), Israel sought its own and did not submit to God’s righteousness. The righteousness of God in 10:3a is about God’s covenant faithfulness to accomplish his promises given to Abraham. Israel resisted the faithfulness of God in the Messiah and preferred its own kind of righteousness: a focus here on Israel’s elective privilege.
That, my friends, is what we mean by Wright’s version of the New Perspective. This text is not about Israel trying to pull itself up to God’s own standard, but is instead about God’s covenant faithfulness in the Messiah and Israel’s rejection of the Messiah.