You refer to "the Pauline perspective" in your comments, that it "prevents commentators from reading Deuteronomy, Ezekiel and Habakkuk as saying the same thing. Really - you need to make the case for that claim. What commentators, precisely? And what grounds are there for claiming that "the Pauline perspective" hinders anyone from seeing those O.T. verses other than "saying the same thing"? Does Paul even hint anywhere that they are not saying the same thing?
I would suggest, by way of an answer, that it is what the entire Bible teaches about the righteousness of God that has to be studied first, before we can understand anything it says about the faith that pleases God, which faith must be based upon, and show forth the righteousness of God.
All the texts you quote are not "simply faithful restatements of one another". They all enshrine biblical principles about the immense importance of how God views righteousness, and how that is linked to saving faith that brings life eternal. One text from the O.T. which Paul quoted (but which you did not mention) is in Romans 3:10-18, showing that nobody is righteous, in and of themselves. He goes on to show that it is the righteousness of God by faith of Jesus Christ that brings free justification to those who have faith in the blood of Christ, who "declare his righteousness" (vss 20-28).
That is the biblical link between God's righteousness and sinners who could never be righteous as is God. He enables unrighteous sinners to live by faith. God is just. He requires those whom he justifies to live by faith and to show forth justice and righteousness in their life of faith. This means that what started to be proclaimed in those OT verses you quoted have the principles contained therein expanded and made clear by Paul's theology. He starts with the righteousness of God and the unpalatable fact that "there is no-one righteous, no, not even one." He gets that from O.T. scriptures in Psalms and Isaiah where those who don't grasp that God is righteous in his divine nature, never understand what true, godly righteousness is. They put the cart before the horse by starting with themselves, supposing they can either be righteous, or earn it by doing the right things.
The OT shockingly declares this in Isaiah 64:4-7. Even God's chosen people were viewed by him as a filthy rag when they tried to do 'righteousness'. That has to be the starting point for those of us who take the O.T. and the N.T. together, to grasp the whole picture with regard to principles of righteousness. Only God is righteous. He cannot be anything else, for that is his very nature. We are all unrighteous no matter how hard we might try not to be. Only God can deal with our unrighteousness and justly remove it because of what Jesus did in that regard.
So, yes, we should understand all those texts you quote to be "the same teaching". That becomes apparent when we make God's righteousness our starting point for grasping the biblical principles about righteousness. The trouble is, many people start with themselves and seek to work out how they can attain to righteousness, yet they have no fear and trembling at the holy righteousness of God! Neither Paul nor the O.T. writers did that. Start with God (as they did), and then comes understanding.