Why did St. Paul treat himself as not under the law?

I consulted the Encyclopedia of Theology, a Concise Sacramentum Mundi edited by Karl Rahner, and found a section headed ‘The Law in Paul’. Herein is found the answer to your question, and I now simply quote the relevant parts (with all emphases mine):

“Paul and Judaism see the law as an independent theological entity: according to the claim of those who submit to it, it is an independent way of salvation, in competition with faith in Jesus Christ.

While Paul leaves untouched the social conception of the law upheld in Judaism, the salvific function of the law becomes a problem for his theology… the law as a way of salvation must be excluded. Salvation consists in the justice of God, i.e., in the fulfilment of the promises made to the patriarchs. Whereas in Judaism justice is achieved by the fulfilment of the law, Paul goes back to the close connection of justice with faith in Gen 15:6 (Abraham) and, with the help of scriptural proofs, is able to show that the promises, and therefore their fulfilment also, are linked to this justification by faith. The law, therefore, as a way of salvation is excluded because those who observe it hope thereby to achieve righteousness, whereas Scripture demonstrates that righteousness is linked with faith. Law and faith (in Jesus Christ) are made exclusive alternatives. Paul goes on to demonstrate that man could never have obtained righteousness by the way of the law, for no one (as Paul postulates) can fulfil it completely (Rom 3:20-30; Gal 3:10; 5:3; cf. 6:13). Hence all men come under the curse which is the penalty threatened in Deut 27:26. But Jesus alone can free man from this curse, for in him is promised the opposite of the curse, the blessing for all peoples already promised to Abraham…

From the time of Adam until Christ the power if sin prevailed, activating the sphere of the sarx, and bringing the sinful works of the flesh to light (von Dulmen), so that the power of sin signified both the sinful act itself and also the state of bondage to sin, while law assisted sin to gain power and life (1 Cor 15:56). For with the advent of the law all became guilty of transgression. In contrast to sin, however, law is only destructive in its effects and function, not by nature. Sin avails of the law in order to kill. The law is the catalyst which reveals the ruinous and hopeless state of man. But when the curse of the law is concentrated upon Jesus he endures the penalty of death, and thereby sets aside the demand of the law. For this reason, and also because the law is no longer used to demonstrate the presence of sin, Christ is the end of the law…

At present the law only has a part to play in bringing out the obstinacy of the unbelieving section of Israel, which opens the way for the Gentiles to be taken into the community (Rom 9:11).” Klaus Berger, pp 828-829 (1981 edition)

That is why Paul did not boast in his Jewish heritage, but boasted in Christ and how faith in this risen saviour had set Paul free from bondage to the law, to be free in Christ. Paul exquisitely shows from the Hebrew scriptures how the law was a shadow pointing to Christ, the reality, so that both Jews and Gentiles putting faith in the finished work of Christ on the cross could have the chains of bondage (to sin and death – sarx ) smashed ( Col 2:13-232). That is why Paul said that he would preach nothing but Christ crucified, as the end of the law (1 Cor 1:23 & 2:2). His point was that all things must lead to the risen Christ, and the law served that purpose for those who did not see the preaching of the cross as ‘foolishness’ but as the power of God unto salvation (1 Cor 1:18, Eph 3:8).

This does not mean that Paul was lawless, or even that he took a light view of law-breaking. On the contrary, he shows in Romans 8:4 that "the righteousness of the law might be fulfilled in us" [Christians who walk after the Spirit.] He exposed as false those who accused him of despising the law (Antinomianism). See Romans 3:31 & 7:7-25.