You still miss the point. In Witness theology and in the New Testament Jesus is mediator of the new covenant. Nothing in what you quoted shows that the 'anointed' or the governing body are mediators. You still confuse Jesus' office as propitiator with his office as mediator. Your quotation does show that Witnesses believe they are the one true religion. They believe that if you leave their religion - since it is the one true religion in their belief - then you reject Jesus' ransom. Nonsense of course. But the WT does not teach that they are mediators.
They teach, and this much is biblical, that Jesus fills two offices. Jesus is a covenant mediator. He is the high priest, who propitiates God in behalf of all men. [1 Jn 2:2; 4:10] You are right in rejecting Witness theology. You are wrong in your terminology, in your definition of Witness doctrine. Witnesses see Christ as the propitiator between God and all men.
Most who object to Witness mediator doctrine think that the mediator represents them to God and covers their sins. That's not the mediatorial office. Jesus' mediation was of the New Covenant. The Bible says this. Witnesses believe that only a few are in that covenant. Even if so, all men benefit by Jesus' office as propitator. Witnesses do not teach that salvation comes by others than Christ. They do teach that they have the one true faith. That's improbable, but other religions see themselves as the one true faith too. They're not exceptional in this.
Broken down into its components, Witness doctrine does not deny salvation to some 'great crowd'. They deny that these are in the New Covenant. As I see it, all Christians are in the New Covenant and Witness doctrine comes not from the Bible but from mid 19th Century teaching by an Anglican and by a Brethren preacher. I oppose their doctrine. But if we wish to refute it, we must be exact in our terminology or we persuade no one but those who already reject their doctrine.