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“any media outlet, no matter how big an empire it is, that is not owned or funded by the state” is not the common definition of “independent media” (it’s not even the definition given in the hyperlinked definition). “Independence” in this context refers to journalistic independence.
“Stuff in my feed I don’t want to see in my feed” is kind of the exact problem the Fediverse set out to solve. Nothing gets “injected” to a feed here so if you are seeing it, it’s a choice to continue to do so.