While you are correct on this, it is a matter of the old covenant vs. the new covenant. The old covenant contained faults...for example, the old covenant could not take away sins, the new covenant does. Hebrews 8:7 says, "For if that first covenant had been faultless, then should no place have been sought for the second."
As Paul wrote, by Jesus being crucified, He was "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" (Colossians 2:14). In my understanding, the Old Testament is a history of God's old covenant with the children of Israel, the New Testament is what we should live by and it tells us of things to come. But like any good book, you always need to read the first half of the book to understand the rest of the book, and regardless of the names of the authors of the chapters, it was divinely inspired by God and the authors were merely "ghost writers", so therefore the whole Bible is of God, as you say.
38But you do not have His word abiding in you, because whom He sent, Him you do not believe. 39You search the Scriptures, for in them you think you have eternal life; and these are they which testify of Me.
44Then He said to them, "These are the words which I spoke to you while I was still with you, that all things must be fulfilled which were written in the Law of Moses and the Prophets and the Psalms concerning Me." 45And He opened their understanding, that they might comprehend the Scriptures.
The New Testament is the fulfillment of the Old. All the Law and the Prophets spoke of and pointed to Christ.