When Rutheford took over the presidency he was faced with several schisms. One on side you had Bible Students loyal to Charles Russell and his teachings. On the Judge's side, you had those Bible Students loyal to the Watchtower and Rutherrford as president. Both Bible Students groups claimed the Trtuh and God's backing and were in many ways indistinguishable.
I believe this was a motivating factor which lead to the Judge rebranding the religion as "Jehovah's Witnesses". It wasn't a mere name change, but a huge doctrinal change. In doing so Rutherford began to apply Bible prophecy that was previously understood as applying to Israel, to the group of Watchtower loyal Bible Students.
The book "The Finished Mystery" was the first released under Rutherford and you can already begin to see this method at work. In the book the 1600 furlongs of Revelation 14:20 are understood to mean the distance from Scranton, Pennsylvania (where the book was written) to the Watchtower Headquarters in Brooklyn. Here the activities of the Society and Watchtower Headquarters are written into Bible prophecy, albeit in a crude, fanciful manner. ("The Finished Mystery" predates most of the schisms and is more generous to Charles Russell then later books would be. )
Rutherford's later books followed this tactic of applying Bible prophecy to the Watchtower. For example the "seven bowls of anger" and "seven trumpets" from Revelation are applied to Watchtower tracts and conventions in the early 1900's. Similarly organization changes in the early 1940's were understood to be the fulfillment of Daniel's prophecy about the Holy Place (Watchtower HQ) being restored. Here By applying prophecy to the Watchtower, he was making his religion the only truth amid various Bible student groups operating in that time. Only the Judge's Watchtower had the backing of God and was being used to fulfill Bible prophecy.
However as this time period fades into history, these teaching become less impactful. Even 1914, the pivotal date used by the Watchtower, is past its sell by. The Organization is not afraid to adjust its understanding of prophecy, but the task of unchaining it from Rutheford's troubled reign might not be an easy one. It would require huge adjustments to their understanding of Bible books like Revelation and Daniel.