As for myself it is very simple. I have no problem at all knowing that the Sun would revolve around the Earth which contains the spot where the Word became Flesh and dwelt amongst us.
The harm of heliocentrism wasn't that it couldn't immediately be proved, but could be in so many ways, ultimately, but that it was used as yet another early wedge against the Faith.
The notion of geological eons followed quickly by Darwinism, was yet another assault.
But it's because many opposed to the Faith of Roman Catholicism, and a lesser number against various Protestantism, that they made logical arguments, or arguments of false logic, based on these things against The Church.
Someone correct me if this is wrong, but it's said that Bellarmine insisted that the earth not only was our point of reference, our 'inertial frame' as it later would be seen, but that the heavens rotated as spheres, and spheres around spheres, etc. around the earth. And that could be disproved. It's not a question of relative motion.
But that the earth remains the center of our universe is without question. That this is the origin of a 'space-time' cone, if you will, is just how it is. From this vantage point, time and space proceed from earth, or at least from the same general area. The heavens may not circle in perfect orbits around earth, or this solar system. But as a point of reference, this IS the point of reference, right where we are now.
There's also something about this place, this planet, that is either missed intentionally or not by certain researchers. This is where life is known. This is where God intervened in the affairs of those made in His Image. This is where the 'action is', in a universe of unimaginably vast nebulae and gigantic or massive stars. It could be that out of all of that, all the factors that had to come into play, did so right here, by God's design. It could be, there's nothing else out there except for digitally colored and enhanced wonders that existed, in many cases, before Adam was even created. And those spectacles are, in a sense, long gone. And we aren't.