The main myth is that: Galileo was obviously right given the evidence at the time, and that the Catholic Church pursued him entirely because they were unwilling to change their view. The evidence instead suggests that the Catholic Church was primarily concerned (oppressively so) about explicit interpretation of scripture, but not so much (as an institution) concerned about matters of speculation about models of the solar system, especially when they were specifically couched as such, and they were also quite willing to adjust their interpretations of scripture if those interpretations were actually falsified by evidence. The problem was the Galileo a) didn't have that evidence (and the evidence he tried to present was wrong: the book that got him in trouble focused on an argument from the tides where he assumed the tides go up and down once a day! He had some other better ideas but the experiments/observations didn't work, sometimes due to sheer bad luck), b) was advocating for a model that absent that evidence was mathematically equivilent to a geocentric model (there were many: only the simplest were eliminated by the new observations of phases of Venus) but more complex (Kepler had the right idea but Gallileo was insistent on the epicycles of circles), and c) made a lot of friends into enemies by generally being an arrogant credit-hoarder, as well as having more unreasonable enemies.

The events that lead up to the house arrest were quiet chaotic as well: an almost comedy of errors meant that Gallileo's book never got a full review in context before publishing and so the implied insult of the Pope was published with Gallileo seemingly believing it had been approved beforehand (seemingly Gallileo had understood the Pope's attitude to be a "just make sure you put a small fig leaf on it until we change our official position" instead of "You don't actually have enough evidence to disprove this other theory, but it's worth writing something up considering both"). The trial also involved a bunch of skullduggery where various documents were fabricated (by parties unknown, though some would have needed to be inside the church) to paint Galileo as if he had directly opposed the church on matters of scripture, which was what ultimately got him in trouble.

So, was the Catholic church extremely overbearing and oppressive in regulating the speech and writings of others? Yes, for sure. But on a quite limited scope: they cared relatively little for scientific speculation. Could Galileo have basically published the same information and not gotten into trouble? Also yes. It took a dedicated conspiracy to try to paint him as attacking the church and him pissing off almost all his friends (including one quite powerful one) for him to get punished.

So, the full story makes the church and pope appear a little more reasonable. It's still unreasonable that they were in the position of "Falsify the current consensus or keep your statements couched in speculative terms", but it's far from "No, shut up with your obvious truth". Similarly, Galileo looks a lot less reasonable: basically insisting he's correct when his own proposal is unwieldy and still only correct in a few more details than the consensus, and it appears mostly by accident because he was unable to actually collect convincing evidence for those details (it's notable that he also didn't really convince many others either. Eventually Kepler's model took over due to mathematical convenience and then more precise measurements provided the evidence Galileo lacked). And there's a whole third part where some even more unreasonable people are trying to take Galileo down for a much smaller slight, and eventially their efforts do contribute to that.