As much as I recognize the compelling facts, especially in relation to the mention of Timothy and his relationship to Paul, and the presumption that the writer was writing from Rome, it is important to note that Timothy had other close companions like Silas. The mere mention of “our brother Timothy” is insufficient in my view, to ground this argument. The greeting that refers to the people of Italy too is insufficient reason. The book does not name its author and there is no unanimity of tradition concerning his identity.
Paul may have ended his letters with warm greetings but this was not unique to him as we see in 3rd John verse 13 “Peace to you, our friends greet you…”, 1 Peter 5:14 “Peace to you all who are in Christ Jesus”. It seemed to be the acceptable way to end epistles.
According to DA Carson and Douglas J Moo in 2005, “The last major defence of the Pauline authorship of the book of Hebrews was written more than half a century ago. Today virtually no one would repeat the effort” They went on to state further that the Greek of Hebrews is more polished than that Paul and the consistent quality of the rhetoric is quite remarkable.