Were the Israelites, or did they enter into it?
If the Law was always to be observed 'gratis' (out of gratitude, responsively) then it never was obligating God to Israel. Apparently in the post-exile, intertestament period that conception came about--that God was obligated to Israel if Israel obeyed perfectly. Thus the pressure to see the Sabbath kept exactly, and the tension about inter-racial marriage, etc., race in general.
As far as I know, that is what the NT means when it says people are 'under the law.' It is post-exile, intertestament Judaism. We know from Gal 3:17 that this doctrinal system no longer believed in an outright promise of Abraham, and had replaced it with the Law, thus the pressure to perform found in Pharisee Judaism. Paul grew up in that.
Nowhere in the Law does it say to write a psalm each day, but David did and God called him a man after his own heart--quite a compliment!
I think what Paul is saying about post-exile, IT Judaism is that it thought of itself as entering an agreement in which it could obligate God. This is what 'establishing its own righteousness' meant. So at the end of Rom 11, after Paul has tried once more to redirect his countrymen from the Law to the mission work of the Gospel, he says the rather pointed 'to whom is God obligated or in debt?' in the doxology.
This view keeps the unity of the Bible: that it always was built around the promised Seed of Gen 3, repeated in ch 12 as the same thing (not to many people, but to one Seed), so that by faith in him people would receive the Spirit of God and work in the mission of God when/after Christ appeared. This promoted the Gospel as wide as possible in Paul's time in addition to God's timed providing of ships to travel upon and a good federal road system and an easy language (koine Greek) for federal communication around the Empire.