Answer

See column 112 about this: https://mikyab.net/%D7%9E%D7%94%D7%99-%D7%A9%D7%99%D7%A8%D7%94-%D7%95-%D7%9C%D7%9E%D7%A9%D7%9E%D7%A2%D7%95%D7%AA%D7%95-%D7%A9%D7%9C-%D7%94%D7%9E%D7%94%D7%9C%D7%9A-%D7%9B%D7%95%D7%9C%D7%95-%D7%98%D7%95%D7%A8-112/
Perhaps you also saw this in one paragraph of my article on the site, “Knowing How to Believe” (which will be brought in a link below).
In any case, the video shows a typical argument in which both sides are wrong.
Indeed, Branover erred badly when he connected this to relativity theory. There is not the slightest connection to relativity theory. This is about Galileo’s principle of relativity, as the critic rightly claims. Galileo already understood that absolute motion cannot be defined, and all motion is relative to some frame of reference.
But Branover was right in his claim that, according to Galileo, the question of who revolves around whom has no meaning. The critic decided that when speaking about rotation one does so relative to the center of mass, but that is an arbitrary decision. Usually this is done relative to the speaker’s location (and that too is of course arbitrary). Therefore, if we are located on Earth, it is reasonable to describe the rotation in our own terms, that is, in a geocentric way. Of course, mathematically the two descriptions are equivalent, and one may choose the easier and more convenient one. And indeed, there is no meaning to the question of who is right and who is wrong.
I would add that all this is on the kinematic plane (describing the motion). On the dynamic plane, physicists tend to use other descriptions in which the question of who revolves around whom can have meaning (the appearance of fictitious forces and the like), but even there this is an arbitrary definition. The kinematics remains unchanged. There is no way to define objectively who revolves around whom. It is a matter of definition. Beyond that, these things are also connected to the debate over Mach’s principle and the bucket problem. See here.
The critic’s claim that according to Branover, Copernicus discovered nothing new, is context-dependent. Copernicus found a more convenient system for the kinematic description (and perhaps also the center of mass of the system), but he did not find a system that is more correct. Note that nobody ever claimed that the geocentric description is more convenient. The Church claimed that it is more correct, and in that it was mistaken. The critic claims that it is less correct, and in that he too is mistaken. Both are equally correct, and the use of either one depends on the purpose and on how efficient each of the two is for that purpose.
The answer to why people laugh at the religious, as the video’s title says, is: somewhat justifiably and somewhat not. Unfortunately, there is ignorance both in the religious world and in the secular one. You can, of course, laugh at everyone, but do so equally. What is surprising is to discover ignorance among professionals, and in my opinion it is the result of an agenda. In Branover’s case, who certainly knows relativity theory and Galileo’s mechanics, the need to impress probably clouded his judgment. He had an agenda to present atheism as outdated and modern science as confirming the religious view. But here that is simply not true. By the same token, the secular critic also displays ignorance, and apparently in his case too it is about an agenda. As I explained, the common use of Copernicus as an argument against the religious view suffers from the same flaw as the religious view being criticized itself.