What purpose is God carrying out during the Millennial Kingdom?

I suggest they do. The church is the kingdom.
When Jesus sent out the apostles, before His sacrifice, He told them to proclaim the kingdom was at hand. During His ministry He describes the kingdom several times. Does His descriptions describe the kingdom you envision on earth or the church?

The theological term that is used to describe the state of the Kingdom at this point is, "Already but not yet". Messiah is already anointed King of the Kingdom of God, but He is not fully in charge of the government here on earth. The Millennial Kingdom is when this is fully realized!

Messiah is already anointed King of the Kingdom of God, but He is not fully in charge of the government here on earth.

The theological term that is used to describe the state of the Kingdom at this point is, "Already but not yet". Messiah is already anointed King of the Kingdom of God, but He is not fully in charge of the government here on earth. The Millennial Kingdom is when this is fully realized!

The church is the greatest kingdom ever and it is eternal.

Once again you are trying to spiritualize things. The Church is NOT the "greatest kingdom". The Church is within the eternal Kingdom of God, which goes well beyond the Church.

The kingdom is within us and, therefore, within the church. Luke 17:21...the kingdom of God is within you...

And that is NOT a substitute for the real, visible tangible Kingdom of God in earth. That is only UNTIL the eternal Kingdom of God is established. This will include both the Church and all the nations on earth under Christ.

There has never been a time when God was not fully in charge.

May I point out that when Jesus and the apostles were declaring the kingdom was at hand, that was before Jesus declared He had been given all authority and also before the church was established.

You didn't address the kingdom Jesus describes. May I suggest the kingdom prophesied in the OT is merely using language they would understand. Describing the greatest kingdom ever. The church is the greatest kingdom ever and it is eternal. In Christ, man now has the forgiveness of sins. It made Jew and Gentile into one new man establishing peace. Through Christ man can appeal to God. The church is sealed by the HS., ect...
When Jesus says, "Therefore I tell you, the kingdom of God will be taken away from you and given to a people producing its fruits" ,or "the kingdom of heaven is like", do you feel Jesus is describing the "earthly" kingdom you envision or the church?

Also, Paul went about preaching the kingdom. Did he preach the future "earthly" kingdom or the church?
25 And now, behold, I know that none of you among whom I have gone about proclaiming the kingdom will see my face again.

8 And he entered the synagogue and for three months spoke boldly, reasoning and persuading them about the kingdom of God.

From morning till evening he expounded to them, testifying to the kingdom of God and trying to convince them about Jesus both from the Law of Moses and from the Prophets.

2 Cor 4:3,4 "And even if our gospel is veiled, it is veiled to those who are perishing, in whose case the god of this world has blinded the minds of the unbelieving so that they might not see the light of the gospel of the glory of Christ, who is the image of God."
Sounds a little more complicated to me!

Simply put, it means,
When a believer truly learns the truth, in the process of learning, understanding the difference between truth and reality is a process of judging the world, silent judgment.

I think you might have stepped into the wrong classroom. Cathartic navel poetry is two doors down.

The New Covenant has already been established - Hebrews 8:7-12
"When He said, “A new covenant,” He has made the first obsolete. But whatever is becoming obsolete and growing old is [fn]ready to disappear." Hebrews 8:13

So . . . no, The Millennial Kingdom has NOTHING to do with establishing a New Covenant!

Till heaven and earth pass, one jot or one tittle shall in no wise pass from the law until all is fulfilled.

Now we know that what things soever the law says, it says to them who are under the law that every mouth may be stopped, and all the world may become guilty before God.

Before faith we were kept under the law shut up unto the faith that should afterwards be revealed. Wherefore the law was our schoolmaster to bring us to Christ that we might be justified by faith. But after faith, we are no longer under a schoolmaster.

For this cause Jesus is the mediator of the new testament, that by means of death, for the redemption of the transgressions that were under the first testament, they which are called might receive the promise of eternal inheritance.

(This means that both the old and new testaments run concurrently until the great white throne judgement where the lost will be judged by the standard contained in the law including those who are without faith during the Millennium. The new testament has no curse of death, except for a very few sins.)

If any man see his brother sin a sin that is not unto death, he shall ask, and He shall give him life for them that sin not unto death. I do not say that you shall pray for it. All unrighteous is sin and there is a sin not unto death.

Yes, well it's all about the Kingdom of God. The Millennial Kingdom is simply a stage or dispensation of the Kingdom of God. Roman Catholics don't like the idea of the Millennial Kingdom because they believe they are the earthly fulfillment of God's Kingdom, and Reformed believers don't like the idea of the Millennial Kingdom because they believe the church has replaced Israel. So both of these groups tend to spiritualize or allegorize this clearly defined period of redemptive future of the earth. It sounds like you are completely unfamiliar with the doctrine of the Millennial Kingdom -it's not something I'm making up but a widely accepted doctrine around the world.

Good stuff except for 2 covenants. The old has faded away. And Jesus blood cleansed heaven and made it new. And the kingdom of God is renewing the earth, also by His blood. Blessing is no longer tied to the law but in Christ.

In the KJV in Matt, it is refers as a "testament", not covenant there is a difference, Jesus our messiah announced his will just before he died on the cross, he initiated the covenant but the it will be in application when he returns only.

Mat 26:28 For this is my blood of the new testament, which is shed for many for the remission of sins.
Mrk 14:24 And he said unto them, This is my blood of the new testament, which is shed for many.

Blessings.

True, I'm not completely familiar with the MK doctrine. If I don't read about it in scripture, I won't be familiar with it.
Just because something is widely accepted doesn't make it true. It's more likely to make it false.
I'm neither RC nor reformed, just Christian.

So the church is not redemptive?

I'm curious what kingdom do you believe Jesus was describing as He went around telling people that it was at hand.