According to Paul, what was it exactly that most of the Jews had failed to grasp regarding the coming of Jesus?
AP: In Romans 10:5-12 Paul says the Jews should also seek the "righteousness that comes from faith (πίστις)" as opposed to "righteousness that comes from the law (νόμος)." In Rom. 10:6 Paul says that that the law of faith is "to bringing Christ" to the Jew (i.e., for the Jewish people to accept Christ), which, according to him, is easy because the word of faith which they preach is in their hearts" (10:8). Look at the words Paul uses. He says: "the word of faith which we are preaching." Here Paul begins to clarify the meaning of pistis in this context. He says it refers to the act of proclaiming the gospel, something the Jews reject but the Gentiles do not. Then Paul asks for everyone to accept the Word of the gospel. He wants them to believe and trust in it. It is crystal clear that Paul is demanding that his Jewish countrymen to put their trust in the gospel that he preached and, by extension, in Christ. It is also crystal clear that what Paul criticizes his fellow Jews for in Romans 9–11 is their lack of faith not only in God's plans, but also in Christ.
The New Radicals argue that the Apostle is a Second-Temple Jew to the core, a faithful observer of the Law, etc. Is it believable that within the core of his theology, Paul rejected what was considered essential in the Judaism of his time, namely that the Hebrew Scriptures—all of them—pointed to a messianic salvation? The messianic temperature in the first century was high. In fact, Josephus records almost ten messianic figures from the death of Herod the Great to the outbreak of the anti-Roman revolution (A.D. 66-70). With each the people were led into a suicidal war against the Empire. They were confident that once the fight commenced God would help with their messiah and defeat the Gentiles. But what are we really asking here? Are we asking if Paul in Romans is arguing that it is not necessary for Israel to believe in the Messiah? Are we asking if Paul says Israel will be saved without believing in the Messiah and that only the Gentiles have to believe this? I can’t believe it. In Gal. 2:7-8 Paul says that the core of his gospel to the uncircumcised is exactly equal to the gospel of the circumcision proclaimed by Peter. So, what is the core of the Judeo-Christian gospel? Israel must accept Jesus as God’s Messiah. To pretend otherwise is to completely misunderstand Paul. To do so isn’t just to misunderstand 2 Corinthians and Galatians, it is to completely misunderstand Romans. There are many other Pauline texts, not just Rom. 11:26. But here we just focusing on Romans.
CAS: In my opinion, Paul criticizes them for not realizing the times in which they were living: (1) that they were living during the first coming of Christ, (2) that the Gentiles will be incorporated into Israel. Nothing more. The usual interpretation of Romans 10 (which, by the way, I do not think you free yourself from!) is, in my view, inconsistent: Paul speaks only of Christ to the Gentiles, and the supposed lack of faith that he allegedly criticized some of his brethren for later (11:23) deals only with with their lack of loyalty to the God of Israel (that is, if we read this passage in light of 3:2-3). Moreover, the metaphor of athletic competition developed in 9:30ff. is just that: a metaphor. Some of them (i.e., some of God’s chosen ones) have failed to grasp the promises of God beyond the election of Israel (of which the law is only a sign). Nor did they grasp the means by which God will redeem the nations. Others (i.e., those whom God had not chosen) seem to have grasped this point. The argument is a little forced because it remains to be seen exactly how many Gentiles will be persuaded by the gospel. Not only that, but certain Jews, including Paul (who, by the way, always speaks of his Jewishness in the present) seem to have understood what is at stake; that always happened in Israel’s history—some got it, some didn’t. There’s nothing new here!
In any case, says Paul, those who seemed to be at a disadvantage are now actually the beneficiaries. And it's happening right before their very eyes. This is, in my interpretation, the only thing that Paul actually says. Moreover, you cannot forget or overlook what he says in Rom. 15:8-9, 18:
"For I say that Christ has become a servant of Israel . . . to confirm the promises given to the fathers, and to give an opportunity for the Gentiles to glorify God for His mercy. . . . For I will not presume to speak of anything except what Christ has accomplished through me, in order to bring the Gentiles to God" (literally, "to his Covenant").We should probably start reading Paul by paying attention to these verses. And, for that matter, we should be very careful in saying that Paul's message called into question someone’s Jewish identity. First, there was no single form of Judaism in the first century, and there was no single form of Jewish identity (see Neusner, Boccaccini, Boyarin, etc.). Second, the issue of the inclusion of Gentiles into Israel had important precedents (there are actually more in favor of this point than those who question it, as I believe I have shown in my book!). And third, even if Paul had criticized Jewish law (something I do not think he did), all he criticizes, in my estimation and those who defend the “Radical New Perspective,” is the supposed need to apply the Law to the Gentiles. It is obvious that such criticism was internal, not external (the apocalyptic Judaism is not exactly lawless, but represents, from the very beginning, a dissenting trend against the establishment of a post exilic Zadokite priesthood and their laws). Beyond that, just as Mario Saban reminded us recently and which you yourself point out, there are ancient Jewish texts arguing that the law ceases to exist in the messianic era and is, therefore, invalid.