We still have the precedence effect where the brain attenuates later arriving sounds , making the need for a waveguide in a damped room somewhat less important ?
When we see a very smooth off axis curve with good directivity from a speaker - there is a risk that we count the 30 degree off axis curve as important as the on axis response, where in reality, in many rooms the off axis response is much less important and the on axis direct sound from the speaker are dominating the perceived sound quality.
One can always theoretically construct some special cases. If you listen to a loudspeaker in the free field at 30°, only the shape of the 30° frequency response is important. Or the loudspeaker is not aligned to the listener, then the 30° frequency response is responsible for the direct sound. But it makes no sense to discuss such cases.
It seems that you see the precedence effect only in terms of localization.
If the same sound signal arrives at a listener with a time delay from different directions, the listener perceives only the direction of the first arriving sound signal - so this refers only to the localization.
But the brain doesn't "attenuates later arriving sounds" as you said, it only "ignores it" (to some extend) in terms of localization.
The presence of delayed sound alone (e.g. reflected sound) already leads to a timbre change (interference, comb filter effects,...), as well as the direction of the incoming reflection at the listener, since this is influenced by the HRTF - which has an huge impact on timbre.
HRTF for 90°, 0° and 180°:
All in all, we manage quite well, our brain expects sound coming from the side to have a different timbre than the same sound coming from the front.
But if the speaker adds additional timbre changes due to a bad speaker radiation, then this is not good - think we all agree on that.
Here is an example of a single loudspeaker measured in the center of the room (as far away as possible from reflecting surfaces), once the reflection-free on-axis frequency response (yellow FR) and once only the reflections without the direct sound (green FR) at 1.8m (70'') distance:
Even under these conditions, the sound pressure level of the reflected sound is already dominant or equal in a wide frequency range. Only above 6kHz the direct sound is dominant. Thus, the timbre of a loudspeaker is largely determined by the reflected sound.
Floyd Toole shows in his YouTube presentation (27min) a nice example of what happens when you only pay attention to a flat on-axis frequency response.
The reflected sound significantly influences the timbre of the loudspeaker.
My opinion is that the direct sound coming from the speaker , on axis, is the most important for perceived sound quality in many cases .
This sentence may be true under certain circumstances, but generally speaking, a poorly designed loudspeaker with a flat on-axis FR will sound less good in a normal listening room than a poorly designed LS where the on-axis FR tries to compensate somewhat for the mistakes made in the radiation (don't get me wrong, it will never sound perfect).
If the direct sound from a speaker is the most important for perceived sound quality - do we really need waveguides for our tweeters ?
We don't need speaker with WG, only not bad designed speaker with reasonably uniform radiation and more or less flat on-axis response (with WG or without).