Which is exactly the point. Whether the earth rotates depends on your frame of reference. You can pick one where it is, you can pick one where it isn't. Neither is "true". So saying the earth orbits the sun is a choice : it is not a true or false statement, just a statement reflecting an opinion. Occam's razor (as well as general mathematical practice) dictates that we declare this to be not true in that case (note the massive difference between "not true" and "false". Not true means you don't know it to be true AND you don't know it to be false either). But this is only possible in euclidean space, you have to transform space (change the normal/distance function) before you can choose a reference frame where there is rotation.

If you look at it in the original relative space, without an euclidean transformation, there's only 2 ways to see it : you can only pick reference frames where the earth is standing still entirely, or you pick reference frames where it is moving along a straight line. You cannot pick a reference frame in relativistic space that shows ellipsoid movement (like the one predicted by the Newtonian theory which is generally what people mean by "orbit").

So you can't actually pick reference frames to do whatever you want. In order to get something looking like an orbit you have to pick a reference frame where you not only have messed with the point of origin, but also projected the distances between objects to be invariant under the influence of gravity. Which they're not in reality (effectively you're claiming that distances everywhere in the universe are as they are at an earth-sun lagrange point, which is not true). Only by doing that transformation can you get an orbital pattern.

At that point, from a purely mathematical point of view you can analyze what other motions you could get that are equally true as saying it orbits. So you can say, you can put the origin point anywhere you like, moving, accelerating, whatever you want. In addition to that you can arbitrarily change the distance function. So you could make one where the earth spirals into the sun. Or spirals away from it (simply introduce a time component in the distance function). Hell, you could make one where the earth and the moon look like they're bouncing on the surface of the sun that looks like the disc in the discworld (put origin at the center of the sun, distance function is a the normal multiplied by the tangent of an angle that goes from the center of the sun to where the moon was 48 hours ago, and in the z direction all distances are zero). Hell, I bet that even if you demanded the reference frame be euclidean you could still make it look pretty silly (I think my spiraling examples would still be euclidean). Reasons for picking one over the other ? None.

Well, one : if you want to calculate the influence of the laws of physics inside our solar system, having an euclidean reference frame centered on the sun-planets lagrange point (which is not the center of the sun) is pretty useful, as it means you need 11 (number of planets) transformations to calculate the path of a satellite moving through the solar system, whereas a relativistic reference frame would require 12 transformations (imho because they don't change rotation they're simpler though).

Aside from the utilitarian choice, all those statements are simply equally true to saying that "the earth rotates around the sun". Just because we prefer one arbitrary kind of reference frame above others here on earth (ie. euclidean) doesn't make it any more real.

And yes, compared to objects "near infinity" the earth arrives at the same spot once a year, but that doesn't make it's path any less straight.