If no one is saved differently in the NT than in the OT, why did God give a New Covenant in replacement of the Old Covenant?
The New covenant isn’t literally a “replacement” of the Old. In the new covenant, the old is written on the hearts:
Jeremiah 31:33 But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD,
I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people.
Even in the “old” the law was to be in the heart:
Isaiah 51:7 Hearken unto me, ye that know righteousness, the people
in whose heart is my law; fear ye not the reproach of men, neither be ye afraid of their revilings.
If we are saved the same way as the OT folks, why aren’t we sacrificing lambs in the back yard? Why did Jesus have to come?
The lambs were symbolic of the Lamb slain from the foundation of the world:
Revelation 13:8 And all that dwell upon the earth shall worship him, whose names are not written in the book of life of the Lamb slain from the foundation of the world.
So, when He came and died ofr our sins the laws that pertained to His sacrifice passed away (were nailed to the Cross):
Colossians 2:14 Blotting out the handwriting of ordinances that was against us, which was contrary to us, and took it out of the way, nailing it to his cross;
Easy answer: because we aren’t saved in the same way.
It is an old adage that for every complex problem there is an easy and wrong answer. Man has always been saved by faith.
Romans 4:9*¶ Cometh this blessedness then upon the circumcision only, or upon the uncircumcision also? for we say that
faith was reckoned to Abraham for righteousness.
Hebrews 11:7
By faith Noah, being warned of God of things not seen as yet, moved with fear, prepared an ark to the saving of his house; by the which he condemned the world, and
became heir of the righteousness which is by faith.