Before you get answers there is good time to do some reasoning and experiments to get some first hand experience on it.
Following is few experiments you can make to figure out the stuff by yourself. As disclaimer I'm just hobbyist but the stuff is based on pretty simple concepts and experiments so I believe there is no need to have degree in acoustics to be able to figure out what you like and how to strive for the sound, so skip it or read it
Dispersion doesn't matter if there is no room and only one person listening as its only direct sound you hear and you can freely position the speakers so that direct sound is best it can. But now in room, turn your speakers around face to somewhere else than listening spot, how does it sound? Take two steps to any direction and what the sound is now? What speaker radiates all around very much affects the sound perceived by hearing system as long as there is room that reflects sound toward listening spot.
However, our remarkable hearing system tries very hard to remove effects of room from perception so all we need to do is make hearing system job easy for nice non fatiguing sound. I'm no expert on psychoacoustics but one such mechanism is precedence effect, lets linger on that a moment. You can hear precedence effect at work if you have good stereo setup going on. When you are listening at sweet spot you can hear nice phantom image on the center, but move little to side and the phantom center jumps to the closest speaker. Try this with some mono signal fed to both speakers it should be quite clear. Its remarkable how almost nothing seems to come from the further speaker when this happens, hearing system is able to completely remove sound from the further source, it prefers the first arriving one. However, if you feed stereo signal to the speakers you would be able to hear stuff from the far speaker that were panned there, sounds that are not present on the closer speaker. I guess this might not happen if the speakers / setup is not very good, in other words center panned sound radiated by each speaker do not correlate well (1). Anyway, you could try and whip up some listening test for precedence effect in similar vein, or just take my word for it for example sake.
Now, imagine the far speaker is a room reflection. When the reflection sounds largely the same as direct sound, and arrives after some particular time interval after first arriving sound it is masked / ignored by hearing system. To apply this to your question, to speakers in room, the off-axis sound should be as similar to direct sound as possible and perhaps past some delay, so hearing system can filter them out. If not, then hearing system is not filtering them out as effectively and the situation is more or less room affecting the sound. You could also brute force this, have speakers that do not radiate sound anywhere else than to the listening spot, narrow spread. Or treat the room so it doesn't reflect sound to listening spot.
Then there is another aspect, rather easy to perceive, reverberation. If you go to a house with high ceilings / little furniture, or to some other huge space like garage or something, its not very comfortable place to be and its relatively hard to understand conversation due to loud (late) reverberation. This can be simplified to concept of ratio between direct and reverberant sound in room. One can affect this with loudspeaker spread (coverage angle), the narrower the coverage is the less the ratio becomes, direct sound gets greater in relation to reverberation simply because more sound is radiated towards the listener and less towards the room to reverberate around. There is probably some particular ratio that sounds nicest to you, not too wet or dry, nonetheless the ratio is affected by speaker spread. When you have speakers with some coverage angle setup in a room with some reverberation characteristics there is probably a distance where you can hear a distinct perception change. No go to your sweet spot and walk along center normal line between stereo speakers. When your are close enough to speakers the sound seems to become more clear and suddenly, perhaps one step further out and the sound gets veiled or something again, quite clear perceptual change it is. This I believe is when direct / reverberant sound ratio gets over a threshold hearing system is not able to filter the sound similarly anymore. If you don't get the effect, move speakers closer together or try single speaker mono.
Anyway, as said I'm not sure what the mechanisms and features on the above experiments actually are that make the perceptual changes but the experiments are quite easy to do. Now its time to figure out which qualities you prefer, more or less clear sound, more or less defined image and so on. It is quite easy to hear and have preference by the experiments. If you like what you hear when the speakers are at your face but not the sound when you sit in your practical listening position its time to rethink the system (2). Otherwise it was just fun time with the hobby
(1) - use time to make your listening setup good to get proper stereo image, change listening position and speaker position until you feel they are as good as practically possible. If still not good the speaker system you have is not the best for your application. Acoustic treatment?
(2) - The speaker system you have is not the best for your application. Acoustic treatment?
So, what "spread" or off-axis response is good depends on your room acoustic properties like size and furnishing, acoustic treatments, and your (practical) listening setup as well as your personal preference. Room and off-axis sound alter timbre and "stereo image" qualities. If you cannot get any of the above stuff happen in your listening setup and system, perhaps try to remember if you have ever had your favorite sound experience somewhere else and what was it like, how far of you are with your setup? Can you remember that situation, setup, room, speakers, whats the difference? Anyway, text on papers, forums is quite hard to translate into perceived sound so I believe and encourage everyone to listen a lot and try to figure out the stuff this way, by experimenting and paying attention whats going on. Even if Toole wrote something, how would you know if you have implemented it properly or not, whether you have reached good sound or not? What is good sound?
I'm not sure if I understood question in the topic title but there is exactly two main "entities" on a loudspeaker that affect perceived sound: what is radiated into room and how. It is pressure variation in room that reaches the ear. One could have very good drivers, even ideal drivers, but if implemented poorly having suboptimal acoustic radiation (in relation to room) and what you hear is suboptimal. You could have very good acoustic radiation, but if the drivers or electronics distort the sound bad with SPL you want to listen to the result is again suboptimal, even if it was radiated ideally to the room.
Have fun!
