The new covenant is superior because, as aforementioned, the old covenant was flawed and kept the children of Israel in bondage.
The law and the prophets, as aforementioned, were fulfilled after Jerusalem was surrounded by armies:
"And when ye shall see Jerusalem compassed with armies, then know that the desolation thereof is nigh. Then let them which are in Judaea flee to the mountains; and let them which are in the midst of it depart out; and let not them that are in the countries enter thereinto. For these be the days of vengeance, that all things which are written may be fulfilled." -- Luk 21:20-22 KJV
It was at that time the temple and Mosaic economy were destroyed, and the kingdom and priesthood were taken away from the Jewish leadership and Levites and given to the faithful remnant from all tribes, as promised in Exodus 19:5-6:
"Therefore say I unto you [chief priests and Pharisees], The kingdom of God shall be taken from you, and given to a nation bringing forth the fruits thereof. And whosoever shall fall on this stone shall be broken: but on whomsoever it shall fall, it will grind him to powder. And when the chief priests and Pharisees had heard his parables, they perceived that he spake of them." -- Mat 21:43-45 KJV
This was the promise:
'And now if ye will indeed hear my voice, and keep my covenant, ye shall be to me a peculiar people above all nations; for the whole earth is mine. And ye shall be to me a royal priesthood and a holy nation: these words shalt thou speak to the children of Israel.' -- Exo 19:5-6 LXX
This was the fulfillment, as Peter explained to the scattered tribes in Asia Minor:
"But ye [diaspora] are a chosen generation [race], a royal priesthood, an holy nation, a peculiar people; that ye should shew forth the praises of him who hath called you out of darkness into his marvellous light; Which in time past were not a people, but are now the people of God: which had not obtained mercy, but now have obtained mercy." -- 1Pet 2:9-10 KJV
As long as the law was in effect, the priesthood belonged to the Levites. But once Christ died and released Israel from the law, Christ became the high priest, and those directly under him became the royal [kingly] priesthood:
"If therefore perfection were by the Levitical priesthood, (for under it the people received the law,) what further need was there that another priest should rise after the order of Melchisedec, and not be called after the order of Aaron? For the priesthood being changed, there is made of necessity a change also of the law. For he of whom these things are spoken pertaineth to another tribe, of which no man gave attendance at the altar. For it is evident that our Lord sprang out of Juda; of which tribe Moses spake nothing concerning priesthood." -- Heb 7:11-14 KJV
This is John, rejoicing that Christ has made him a king and priest:
"And from Jesus Christ, who is the faithful witness, and the first begotten of the dead, and the prince of the kings of the earth. Unto him that loved us, and washed us from our sins in his own blood, And hath made us kings and priests unto God and his Father; to him be glory and dominion for ever and ever. Amen." -- Rev 1:5-6 KJV
That could not happen under the law.
What is the context? Earlier in that chapter it is written:
"Now we know that what things soever the law saith, it saith to them who are under the law: that every mouth may be stopped, and all the world may become guilty before God. Therefore by the deeds of the law there shall no flesh be justified in his sight: for by the law is the knowledge of sin." -- Rom 3:19-20 KJV
Now, pay attention to the next verse:
"But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets;" -- Rom 3:21 KJV
And later, this:
"Where is boasting then? It is excluded. By what law? of works? Nay: but by the law of faith. Therefore we conclude that a man is justified by faith without the deeds of the law." -- Rom 3:27-28 KJV
Now to your verse, in context:
"Seeing it is one God, which shall justify the circumcision by faith, and uncircumcision through faith. Do we then make void the law through faith? God forbid: yea, we establish the law." -- Rom 3:30-31 KJV
Therefore, the law being established was the law of faith.
What is the context? That passage begins with this:
"For sin shall not have dominion over you: for ye are not under the law, but under grace. What then? shall we sin, because we are not under the law, but under grace? God forbid." -- Rom 6:14-15 KJV
I am not sure what your point is.
Again, let us present some context:
"But now we are delivered from the law, that being dead wherein we were held; that we should serve in newness of spirit, and not in the oldness of the letter." -- Rom 7:6 KJV
"There is therefore now no condemnation to them which are in Christ Jesus, who walk not after the flesh, but after the Spirit. For the law of the Spirit of life in Christ Jesus hath made me free from the law of sin and death. For what the law could not do, in that it was weak through the flesh, God sending his own Son in the likeness of sinful flesh, and for sin, condemned sin in the flesh: That the righteousness of the law might be fulfilled in us, who walk not after the flesh, but after the Spirit." -- Rom 8:1-4 KJV
This is very important:
"For they that are after the flesh do mind the things of the flesh; but they that are after the Spirit the things of the Spirit. For to be carnally minded is death; but to be spiritually minded is life and peace." -- Rom 8:5-6 KJV
Again, Christ died to release Israel from the law:
"Wherefore then serveth the law? It was added because of transgressions, till the seed should come to whom the promise was made; and it was ordained by angels in the hand of a mediator." -- Gal 3:19 KJV
"For he is our peace, who hath made both [Jews and Gentiles] one, and hath broken down the middle wall of partition between us; Having abolished in his flesh the enmity, even the law of commandments contained in ordinances; for to make in himself of twain one new man, so making peace;" -- Eph 2:14-15 KJV
I cannot stress this next passage enough to those who believe the law of Moses is still in effect:
"But before faith came, we were kept under the law, shut up unto the faith which should afterwards be revealed. Wherefore the law was our schoolmaster to bring us unto Christ, that we might be justified by faith. But after that faith is come, we are no longer under a schoolmaster." -- Gal 3:23-25 KJV
Rev