A waveguide is a duct for conducting waves efficiently, a transmission line in acoustics. That was how I (half)learned about it.
The association of acoustical waveguides and horns comes from Earl Geddes wanting to distinguish his method from that outlined by Webster's horn equation.
A horn by its most useful property is an acoustic load transformer. It extends the effective radiating area of the driver.
So we go back to the primary drawback of a horn. A horn needs to be large to extend its useful bandwidth.
There are finer points of increased HF nonlinear distortion and nonlinear compression, but these are not too relevant beyond PA use at extremely high SPL.
Regarding higher order modes, or reflections within the horn, my understanding is that they are present in all horns.
Among the exact propagation solutions, there exist only 3 one-parameter horns for which we can totally avoid the excitation of modes besides the fundamental mode. For a spherical wave expansion, this is exclusively the conical horn excited by a spherical wavefront. This should have come up several times in discussing KEF's "ideal" compression horn design and their metamaterial absorber.
The OS horn is optimal for transforming plane waves into spherical waves, but there are necessarily modes excited beyond the fundamental. Geddes didn't want to bother with the problem of figuring out how to create a spherical wavefront in reality and addressed the remaining modes one at a time.
As to whether HOMs are necessarily bad, that depends on how you view things like the diffraction slots in the Image Control Waveguide or Seas DXT tweeter. You can extend that to dual compression drivers, for which the negative effects seem more apparent. The cross chamber resonances create aberrations in the magnitude response that a single compression driver could eliminate.
In recent interview Earl Geddes at 01:03:40
52 minutes.
Although what I think Geddes is taking particular interest in is the modification of the OS profile into a finite horn. The OS solution of wave propagation assumes an infinite length horn. The OS-SE profile is finite, closed-form, and C^infinity. The discontinuity at the horn mouth has caused some problems for horns, particularly the axisymmetric OS horn.
Spatially independent filter being applied to spatially dependent response functions.
To consider a different HOM problem, this is like saying EQ fixes room reflections from the sidewalls. The propagation of the wavefront is the problem, so the response function correction is only applicable within some limited region of space.