There's massive amounts of info on diyaudio about horn colorations, some huge threads if anyone wants to spend hours and hours reading about it.
It's too much and probably too boring and off topic to go on about, but the one thing I'd say is if it doesn't bother you I think the brain tends to disregard it after a while, kind of like how you get used to smells after a short time. I think that's how some live with horns that have more coloration, but if it bothers you from the start it's probably not going to happen for you. Some colorations are more "benign" than others.
IMO all horns have some amount of coloration, it's unavoidable. This is why horn material is so important, it's audible. That's why Cessaro has composite horns filled with some sort of damping fluid, they go through extremes to reduce it. Wood tends to sound decent, and composite materials like fiberglass seem to work well too, but they do sound different. I have composite horns, and when I coated them with a few cans of plastidip to damp them it was audible.
However, boxes are audible too, some worse than others... the baffle shape effects the response of the speaker to a large degree, modern speaker simulators can model this easily, it's in all the books, etc... and the material the box is made out of is audible (YG, Magico, Wilson and others may claim different, but they go to extremes to deal with it). Stats have frame and membrane audibility issues. So, I think it's not only horns that have these types of design considerations.