I agree, it can be audible as horn designs must compromise ideal shape to save on diameter. This can be avoided by using many horns, each covering a frequency range well within the horn's bandwidth to minimize diffraction around the mouth of the horn.
Large multi-way horns are not feasible for most rooms, budgets, time or interest from the owner, etc.
Smaller horns also make compromises to achieve smaller form, such as using innately compromised profiles as well as truncating the horn before it has a full round-over. Where you hear the "honk" depends on the diameter of the horn.
So, there IS audible diffraction around the edge of a horn to some degree in most commercial designs.
OTOH, the most ideal profile (LeCleach) is about DOUBLE the diameter of some commonly used profiles when allowed a full roundover. This introduces quite a few design constraints that may not fit with the overall goals of the designer. OT3rdH, I believe a design using a LeCleach horn profile with roundover can work out very well for some goals and applications... and it does eliminate that issue!
Also, just like cabinet material in a box speaker, horn material is audible. There are some horns made of injection molded plastic, idk, maybe this audible too?
EDIT: I also want to clarify "honk" is a very complex subject with many potential causes, you can read about this for days on diyaudio, search for "HOMs". Honk can come from other things too, but I do think that many non-ideal profiles and lack of roundover are audible.