Justification, Righteousness and Gal 2:17 and 21

Galatians is one of the main text over which the battle for justification has been, and is again being fought.

The word 'justified' in v. 17 comes from the same root word which at other places in English versions is translated 'righteousness. For example, 'righteousness' in v. 21 is from the same root as 'justified' in v. 17. Justified sounds quite different from righteousness, but the two are very closely related. I think the idea behind these words, or this word group if you prefer, is being made right with God.
So v. 17 would become something like (and this is very inelegant English):
'But if, in our endeavour to be made right with God in Christ, we too were found to be sinners, is Christ then a servant of sin? Certainly not!'
And v. 21:
'I do not nullify the grace of God, for if being right with God were through the law, then Christ died for no purpose.'

Two comments arise here.
1) It is God who is the object of justification/righteousness. We find ourselves in a condition of not being right with God and need to be made right with him. Righteousness or justification is not something we need to live in this world as godless humans, but if we desire to live together with God we must be first made right with him. This is the work, which God achieves for us, that is described by these two terms righteousness/justification.
2) These verses Gal 2:17 and 21 make it clear that our being justified or made righteous involves Christ. The 'in Christ' from v. 17 tells us that only united together with Christ can we be made right with God. And v. 21 adds the specific note that the death of Christ, rather than the law, achieves our being right with God, otherwise Christ's death is pointless. It is as we understand Christ's death to achieve our being made right with God that we see this work as not ours but God's, in Christ or through Christ, on our behalf.

These verses say nothing about a transfer of Christ's righteousness to us, nor offer any idea as to how the death of Christ might have achieved this great blessing for us. It is always good, indeed I would say vital, to stop where the bible stops and say nothing more. Now it may be that other passages add to Gal 2, but this post is on these verses not an attempt to offer a full description of justification and righteousness.

Does it make any difference to connect the words 'justified/justification' with the words 'righteous/righteousness'? What does it mean for us and our lives that God has achieved our justification/righteousness through the death of Christ?