As I suspected, there’s consternation about Cosmos’s highlighting of Giordano Bruno, and not all of it is coming from religious apologists.
In Cosmos, Tyson does carefully say that Bruno was not a scientist, and instead describes that picture of infinite worlds as a “guess.” But Bruno was not guessing. He was advancing his own, heretical theology, which goes a long way to understanding the real reason that he was burned at the stake.
The Roman Inquisition listed eight charges against Bruno. His belief in the plurality of worlds was just one. The others involved denying the divinity of Jesus, denying the virgin birth, denying transubstantiation, practicing magic, and believing that animals and objects (including the Earth) possessed souls. You could fairly call Bruno a martyr to the cause of religious freedom, but his cosmic worldview was neither a deduction nor a guess. It was a philosophical corollary of his heterodox belief that God and souls filled all of the universe.
From what I’ve read, the idea of infinite space goes all the way back to the ancient Greeks. Nicolas of Cusa had the insight that other stars might be worlds. What Bruno appears to have added was that they were suns with worlds circling them, where other beings might live.
Was he the first person to think of this? Probably not. And his writing were definitely not scientific. But they were among the earliest on these ideas. And while, as Powell notes, it wasn’t the main basis of his condemnation, one of the charges was “claiming the existence of a plurality of worlds and their eternity.” If the church didn’t care about that part, they would have omitted it.
I think the main point of the Bruno sequence on Cosmos was simply to show that repression of ideas is harmful. Reading more into it, such as it being some kind of anti-religious statement, ignores Neal deGrasse Tyson’s public positions on these matters, or that many religious people deplore inquisitions as much as secular people.