I'm not convinced the Streisand Effect is actually real.

This is a bizarre position to take. The effect does not constitute a claim that all else being equal attempts to suppress information are negatively successful. Instead it describes those cases where information is published more widely due to the suppression attempt. This clearly happens sometimes. The Wikipedia article gives plenty of unambiguous examples.

In April 2007, an attempt at blocking an Advanced Access Content System (AACS) key from being disseminated on Digg caused an uproar when cease-and-desist letters demanded the code be removed from several high-profile websites. This led to the key's proliferation across other sites and chat rooms in various formats, with one commentator describing it as having become "the most famous number on the internet". Within a month, the key had been reprinted on over 280,000 pages, printed on T-shirts and tattoos, and had appeared on YouTube in a song played over 45,000 times.

It would be absurd to believe that the number in question would have been made into T-shirts, tattoos and a popular YouTube song no attempt was made to suppress it. That doesn't mean (or require) that in other cases (and particularly in other cases where the technological and social environment was completely different) that sometimes powerful figures are successful in suppressing information.