Centralized control of media is a huge problem, and yes gradually deregulating it and getting rid of the guard rails had a lot to do with that. Capitalism is just dangerous in general, you need to have strong safeguards for it once it gets too big. I'm not really in agreement that that has anything to do with censoring Nazis in leftist spaces though.
For one thing, the actual literal Nazis are generally so unpopular that they get booted regardless of any kind of philosophy conversation. A lot of the time it seems like those "deplatforming" efforts seem like they're targeted at someone who's not really a Nazi, but just someone who is trying to publish some leftist thing on Substack (which is Officially Problematic (TM) because Richard Spenser is there), or said something which a very cursory reading could consider as transphobic, or something like that.