On the moon, specifically, you have to keep in mind that the European astronomical tradition up to (and even beyond, a bit) Galileo was heavily rooted in a Platonic-Aristotelian approach to the heavens. Which is to say, they took for granted that the most perfect shapes were spheres, and that everything from the moon upwards was in the celestial heavens, and was thus perfect. So everything moved in spheres, and everything was made of spheres.

There is a long line of logic that gets you to this conclusion; what is important is that this is what they believed had to be true. It just made sense, and this idea by Galileo's time was well-integrated into Christian conceptions of God creating a perfect heavens. (One should note that there were some in the Ancient world who thought the Moon had a rougher surface, but their views didn't really carry the day.)

So if you are starting from the idea that the Sun and the Moon are these perfect heavenly objects, how do you reconcile the Moon's variation? What is interesting is that there really isn't a lot of evidence of people trying to do that. It was not treated as a major "problem" to be solved, even though of course everyone knew of the "man in the moon" and that there were variations in appearance. The standard answer was that it was like a crystal with variations in density and internal shadows, and that is what the shapes were. Somewhat amazingly, pre-telescope almost nobody apparently tried to draw the shapes of the Moon with any accuracy. (Da Vinci is one of the only pre-17th century examples we have of this, and he believed the Moon was covered in water.)

It is one of those curious things that pops up all the time in the history of science: the worldview you have determines the problems you think need to be solved. The philosopher Thomas Kuhn considered this sort of aggregate set of ideas about how the world worked and what needed to be explained as a "paradigm," and interestingly the Aristotelian paradigm really didn't give a lot of attention to the surface of the Moon. Galileo reframed the argument with his observations and interpretations of them, essentially arguing that the surface of the Moon was really important, since it directly went against the Aristotelian approach.

By itself this wouldn't have been much of an attack, by combined with his other observations (Jupiter having moons, there being more stars than people realized), Siderius Nuncius adds up to a major attack on the assumptions that underlaid the Aristotelian view. So too the observation that the Sun had spots. Though all of these could somewhat be brushed away as non-challenging to the essential mechanics of the heavens; they're philosophically unnerving, but that's about it. But Galileo's observation of the phases of Venus was totally incompatible with a Ptolemaic system, necessitating (and causing) major change.

The kind of "problem" that the Aristotelians recognized was not the surface of the moon, but things like the retrograde motion of Mars, and explaining all heavenly motion as spheres. That is what most of the astronomy prior to Galileo concerned itself with: finding the right equations/model for thinking about a spherical geocentric heavens that "added up" to the observations they saw. This involved tweaking lots of variables in very complex arrangements, and sometimes even coming up with new ways of doing math to even try these new equations out.

Galileo's approach was a little out of left-field as a result — the kind of observational astronomy he did was just something very different from what the majority of the astronomical profession was doing at that time (if they were observing, it was about getting data to compute orbits). It was not so much Galileo saying, "this specific idea about the Moon is wrong" so much as him saying, "your entire approach to astronomy is wrong, and the craters on the Moon are just one of several reasons to think that."

u/restricteddata has an overview of the whole matter here

https://www.reddit.com/r/AskHistorians/comments/9bbppd/was_galileos_trial_his_own_fault_can_some/

I don't know an exact answer to your question except that I don't think this is true - note in the answer that Galileo's observational evidence was generally accepted by the church. His observations of the phases of Venus prompted them to actually change from the purely simple geocentric Ptolemaic model to the Tychonian model. The church was interested in his observations and not entirely opposed to new ideas, but his heliocentric model was a bit of a stretch for them, especially since he couldn't adequately explain some of the ramifications. He primarily got in trouble for writing his Dialogue book in a way that painted the church, and pope, as idiots.