The problem of physicallizing the Mosiac covenant is that the circumcision of heart is not a new concept introduced by the "New Covenant". Moses urged the Israelite to circumcise their heart so that they have the internal reality corresponding with the external sign:
Deut 10:16 Circumcise therefore the foreskin of your heart, and be no longer stubborn.
This is in the Mosaic covenant, isn't it?
Yes, it is. But the Lord circumcising the heart is a promise made that will occur after the curses of the Mosaic Covenant take place and the people are gathered back (obvious reference to times of Messiah).
Deuteronomy 30:5-6 - "Then the LORD your God will bring you to the land which your fathers possessed, and you shall possess it. He will prosper you and multiply you more than your fathers. 6 "And the LORD your God will circumcise your heart and the heart of your descendants, to love the LORD your God with all your heart and with all your soul, that you may live.
There are prominent typological elements in the Mosiac covenant, but to say that the Mosaic covenant is purely external is to beg the question: "How are those under the Mosaic covenant saved?"
I never said the Mosaic Covenant required that there were no internal aspects. BUT, the overwhelming characteristic of the Mosaic Covenant is that it is external. There are people under the Mosaic Covenant that were not saved. They had no internal relation to God. They were physically in the covenant, but spiritually dead.
In answer to your question, they are saved the same way they were saved before the Mosaic covenant and after the Mosaic covenant. They are justified by faith, just like Abraham.
And what are the functions of the animal sacrifice in the Mosaic covenant? Are there forgiveness of sin in the Mosaic covenant?
Temporary covering until the blood of the Lamb of God was shed. As well as typological of the future sacrifice. As well as to teach that a holy God demands holiness in His people. The forgiveness of sin was always based on the blood of Christ, not on the sacrificial system of Moses.
Hebrews 10:4 - For it is not possible that the blood of bulls and goats could take away sins.
Which lead us to: there are both external and internal elements in both the Abrahamic and Mosaic covenant.
Agreed and never denied. The New Covenant is the one that, in the administration of the Abrahamic Covenant, makes a major advance. The New Covenant consists of the law being written on the heart and the Spirit being put in the hearts. This cannot happen to an unregenerate person. It can only happen to the regenerate.
I hope I don't sound too dispensational here!

I'm still trying to fight off years of bad teaching.
I do believe that regeneration took place under the Old Covenant. That is the only way anyone could be saved by faith. This is the point of John 3. Nicodemus should have understood this. But the difference appears that the New Covenant is made with the remnant that is left of the "house of Judah" and the "house of Israel." The remnant would be the elect. There would not be any non-elect in the remnant because it is the "remnant that will be saved."
Joel 2:32 - And it shall come to pass That whoever calls on the name of the LORD Shall be saved. For in Mount Zion and in Jerusalem there shall be deliverance, As the LORD has said, Among the remnant whom the LORD calls.
Romans 9:27 - Isaiah also cries out concerning Israel: "Though the number of the children of Israel be as the sand of the sea, The remnant will be saved.
And while many can do the external things, it does not mean they are really part of the covenant.
Again, you are confusing the covenant sign and the things it signified.
But you seem to be confusing external actions with being part of the covenant.
When was the last time you see visibly someone baptized by the Holy Spirit? Is it visible? Did you see the doves? Are there any visible sign of the New Covenant? If the signs are not visible, can they still be called covenant sign? Where do you find scriptural warrant that the New Covenant is completely new in the following ways: 1. It has no visible covenant sign. 2. We don't know who are in the New Covenant and who are not because it only belongs to the elect. 3. It is completely internal.
Good points at the end. I thought of some of that as I was typing. I do believe that baptism is a sign of the covenant. You could add the Lord's Supper to that. My point is that I disagree with your assertion that anyone who performs the sign is automatically in covenant with God.
Abraham was not in covenant with God because he circumcised himself. He was in covenant with God because God called him in Genesis 12 and made a unilateral covenant with him. He performed the sign of the covenant in Genesis 17 because he was already in covenant, not in order to get into covenant with God.
A person does not become part of the covenant by being baptized either (or taking the Lord's Supper). A person performs the sign of the covenant (baptism and Lord's Supper) because they are already in covenant.
Any heathen Gentile in Egypt could have circumcised themselves during the time of Abraham (similar to the people of Shechem in Genesis 34) and not been in covenant with God. In the same vein, anyone can be baptized and partake of the Lord's Supper and be lying about being in covenant with God. Doing the sign does not necessarily mean there is a reality.