Please show where the NT writers said Mary was sinless.
You're starting from false premises.
The first false premise is that the NT writers were mindful that the Church would have to convince several generations of skeptics who would not arrive on the scene until a millenium and a half later.
The second false premise is that the writers of the NT took pains to include absolutely every last detail about Christian dogma, liturgy, prayer and worship in the NT.
The third false premise is that the only source of Christian teaching is a book.
Jesus did not commission a book. He commissioned men....real, live human beings! Some of those men whom he commissioned subsequently recorded some of the events in Jesus' life, together with some of his teachings. Likewise, these men commissioned other men who evangelized in various parts of what is now the Middle East, Europe and Asia. Some of their writings to the fledgling Christian communities were likewise preserved in a book.
However, it should be noted that these men had lives apart from any book. Their evangelization did not depend on the New Testament because it didn't exist. For instance, when Paul and Barnabas got into a dispute in Antioch over circumcision, they could not say......"OK....what does it say in the New Testament" (as you just did). Instead, they returned to Jerusalem to consult with "the apostles and priests" as we're told. That's how the Church functioned and how disputes were resolved. No book.....no Bible reading.....just the elders of the Church.
This is the model of the Church which has come down to us to this day. The Bible does not give us the Church. That's bass ackwards.
Likewise, the Christians at Corinth and Ephesus did not receive Paul's letters as readings from the Bible. They received them as letters from the man who had evangelized them.
So in response to your question about where the Bible mentions the Rosary; it doesn't (although the events in the Rosary are described in the Bible).
But so what? It's an issue only for those who embrace the false premises I've listed above.
If things were left out that we needed to know, then it would not be possible for Paul to say that it was profitable and could make the man of God COMPLETE.
2 Timothy 3:14-17 But as for you, continue in what you have learned and have firmly believed, knowing from whom you learned it and how from childhood you have been acquainted with the sacred writings, which are able to make you wise for salvation through faith in Christ Jesus. All Scripture is breathed out by God and profitable for teaching, for reproof, for correction, and for training in righteousness, that the man of God may be complete, equipped for every good work.
Or did God lie when He inspired Paul to write that Scripture was able to make the man of God COMPLETE?
Too much of a target rich post of yours to address each point. Others can chime in. I will address this one.
God moved men through the HOLY SPIRIT to record His words.
Men did not pick and choose what to record. The Holy Spirit moved them, inspiring the words that were used. God made this decision. These are not the "books" of men. They are the words of God. They come from Him.
They are inspired, infallible, authoritative and the direct revelation of truth that all other practices and teachings are judged by.