If God is perfect; why was the first covenant faulty?

For if that first covenant had been faultless, then should no place have been sought for the second. For finding fault with them, he saith, Behold, the days come, saith the Lord, when I will make a new covenant with the house of Israel and with the house of Judah: Not according to the covenant that I made with their fathers in the day when I took them by the hand to lead them out of the land of Egypt; because they continued not in my covenant, and I regarded them not, saith the Lord.

For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people: And they shall not teach every man his neighbour, and every man his brother, saying, Know the Lord: for all shall know me, from the least to the greatest.
— Hebrews 8:7–11 (KJV)

The "fault" with the first covenant was that it failed. One of the parties (the Israelites) failed to live up to their part of the agreement:

• … fault with them …
• … they continued not in my covenant …

The new covenant will not be one of physical obedience, but one of spiritual obedience. People will have God's spirit to individually guide them.

• … I will put my laws into their mind and write them in their hearts …
• … for all shall know me …

This is clearer in the NLT version:

If the first covenant had been faultless, there would have been no need for a second covenant to replace it. But when God found fault with the people, he said: “The day is coming, says the LORD, when I will make a new covenant with the people of Israel and Judah. This covenant will not be like the one I made with their ancestors when I took them by the hand and led them out of the land of Egypt. They did not remain faithful to my covenant, so I turned my back on them, says the LORD.

But this is the new covenant I will make with the people of Israel on that day, says the LORD: I will put my laws in their minds, and I will write them on their hearts. I will be their God, and they will be my people. And they will not need to teach their neighbors, nor will they need to teach their relatives, saying, ‘You should know the LORD.’ For everyone, from the least to the greatest, will know me already.

The Israelites were "the chosen people", but they were chosen not as a special people that is somehow better or more favoured than everyone else. They were chosen to set an example to the world, to show that when the leaders and the people followed God's Law, the nation prospered, and that when society turned to idolatry and other gods, the nation suffered.
(This observation was the reason that the Pharisees added to God's Laws, so that obeying their version of the law would guarantee that God's Law is also followed, and hence so that Israel would prosper.)

During this current age, only a relatively small number of people will become part of God's new covenant. These "saints" or "elect" will become the leaders and teachers during the Millennium, when everyone will be called to receive God's salvation and to have his Law "in their minds, and [written] on their hearts".

But people must freely and individually choose this process. Having ancient Israel as an example helps people to see and understand how essential God's laws are, and what happens when they are disobeyed.

See my (and other's) answers to:

• How do Christians that believe in creatio ex nihilo answer the question of why human beings are not created with a perfect character from the outset?
• What is the basis for the belief "It is possible to have free-will and be incapable of sinning, but it is not possible to be created in that state"?

God warned humankind from the beginning that for naturally created creatures to attempt to live ('life' being more than a physical, animal, existence) by his own strength, his own ability, his own mental apprehensions and, above all, by his own (supposed) inherent 'rightness' - was a fatal mistake.

Humankind having failed utterly, by their own disobedience to God's word, by the deceptive nature of a spirit other than God's spirit, and by their natural weakness, God instituted a covenant - an agreement between two parties - that highlighted the law that humankind had actually chosen to live by.

The failure of humanity was brought into sharp focus by the full revelation (at Sinai) of what the law really was and how far-reaching it was and how deeply it reached and how beyond the ability of flesh and blood it truly was.

What was the answer ?

A new birth ; a new humanity ; another covenant altogether.

These things are evident in the gospel and very evident upon a full reading of the epistle to the Hebrews.

It was impossible to 'start' with the 'covenant of Christ', sic, (the bible does not mention these precise words in this way) for created mankind is not capable of receiving it. Ye must, says Jesus of Nazareth, be born again.

There is no other way.

We must repent - first - and receive truth and believe. Only thus do we enter into a relationship with God, in Christ. And thus partake of the covenant that is between the Father and the Son.

OP: Surely God knew that the first could not be kept by man. Why didn’t He just start with the covenant of Christ?

While God knew it, man didn't. Man had to be taught our inability to be righteous. Still today, some believe they can successfully work to merit right standing with God.

Paul put it this way.

And be found in him, not having mine own righteousness, which is of the law, but that which is through the faith of Christ, the righteousness which is of God by faith: Phil 3:9

The Mosaic Covenant was an IF/THEN. If you can do XYZ, then I (God) will do ABC.

And it [Mosaic Law] shall be our righteousness, if we observe to do all these commandments before the LORD our God, as he hath commanded us. Deut 6:25

The Law was to show everyone that we can't do it. We all fail.

But the scripture hath concluded all under sin, that the promise by faith of Jesus Christ might be given to them that believe. Wherefore the law was our schoolmaster to bring us unto Christ, that we might be justified by faith. Gal 3:22, 24

So, God didn't fail. He showed we were all sinners in need of a Saviour.

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.  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. Because the carnal mind is enmity against God: for it is not subject to the law of God, neither indeed can be. So then they that are in the flesh cannot please God. - Romans 8:1-8

It is important to remember that the law was given to an already corrupted humanity. Adam had already fallen, had already willfully chosen to usurp and take for himself God's lone prerogative in determining good from evil; for this was the temptation:

... your eyes shall be opened, and ye shall be as gods, knowing good and evil. - Genesis 3:5b 

This disposition of sin is part and parcel of each individual human and humanity in general and it's wages are death. In Adam all die. Retaining this intrinsic disposition all die. How to convince a people who are intrinsically predisposed to discern between good and evil for themselves that they do not actually possess such an ability or right and, therefore, cannot actually ever be righteous?

The first step in getting someone to accept a rescue is convincing them that they need rescuing. Any addict in recovery knows this to be true. We are addicted to our sinful disposition. We love darkness rather than light because our deeds are evil. Our sinful nature would have us believe that we are just fine. The Law is, if you will, God's intervention not to rescue us itself but to demonstrate that we need rescue (unless one is bold (and foolish) enough to claim absolutely flawless obedience):

For whosoever shall keep the whole law, and yet offend in one point, he is guilty of all. - James 2:10

The law (old covenant) was given to a humanity at enmity with God and intended to drive us to that faith that admits we are sinners whose only hope is the mercy and grace of God which was ultimately and fully revealed in Christ:

Is the law then against the promises of God? God forbid: for if there had been a law given which could have given life, verily righteousness should have been by the law.  But the scripture hath concluded all under sin, that the promise by faith of Jesus Christ might be given to them that believe. 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. - Galatians 3:21-24

The righteousness that a person needs does not come by the law but the law came to show us that we do not have nor can we attain to the righteousness that is needed. Once that was made clear, then Christ came to provide.

Those who still operate under the assumption that they can attain or maintain righteousness through law-keeping have no use for Christ. They are in rejection of or blind to the purpose of the law and, until that veil is lifted, cannot be saved because they do not perceive their need:

Seeing then that we have such hope, we use great plainness of speech: And not as Moses, which put a vail over his face, that the children of Israel could not stedfastly look to the end of that which is abolished: But their minds were blinded: for until this day remaineth the same vail untaken away in the reading of the old testament; which vail is done away in Christ. But even unto this day, when Moses is read, the vail is upon their heart. Nevertheless when it shall turn to the Lord, the vail shall be taken away. - 2 Corinthians 3:12-16

Generally speaking, if the law had not come first no one would have perception of needing salvation.