So there was a potential and a promise I think that theosophy could have acted as an important go between but it squandered it's opportunities..
Olcott favored Theravadin Buddhism others preferred Tibetan forms .. there was a link to the Arya Samaj and so on.. Besant championed a free India and is still venerated for her work.
Today the world is much more unified and cosmopolitan. The need for a go between is really no longer there.
The movement also splintered and various leaders took off in their own directions .. so pretty much from the start there was infighting and conflicts that have had their affects to this day.
My own take on any "links" between Nazism and Theosophy is that by the time Nazism was growing it's probably true to say there were some German occultists who were behind it but they did not represent most theosophists .. as the movement was already badly splintered.
I see though theosophy as movement that was influenced early on by spiritualism that was very widespread.. and there were antecedents in Swedenborgianism and before that with Rosicrucian ideas..
Jacob Boehme the shoe repairman has a special place though to me and was known as the German Theosopher ... Also Jacobius Comenius has a special place and a crucial one and will be sourced and thought about for a long time in the future..