To me the main takeaway of this thread is reinforcement of something I've longed believed: any time there are directivity flaws, it's likely to be heard as different things by different people in different rooms. To me optimizing directivity is not just about creating the best possible sound, but about creating the most
consistent sound. That's actually one of the most important reasons for optimizing vertical directivity IMO.
I definitely think that the tweeter directivity likely contributes strongly to your impressions reason for your impressions, although did you try EQing the speaker yet? I'm guessing it won't help, but still curious!
Although it's likely the beaming tweeter is a problem for you and your setup, the reason I'm not 100% convinced that beaming is a problem for
most people is that (as you mention) at higher frequencies, we just hear more of the direct sound especially with regards to tonality.
Of course the off-axis response still plays a role. Depends really on the size of the room too. Here's the graph from Toole's book (Section 10.3) about how perception of on-axis vs off-axis varies in a typical room, for those who haven't seen it:
(note that the middle portion is direct+early reflections, not
just early reflections).
As you mention, it's hard to know "how much" the balance of direct and reflected sound changes. But in general, above 5-6 khz it seems, most of what we hear is the direct sound, which is why I don't usually consider a beaming tweeter to be that much of an issue for
tonal balance . However it does affect spatial presentation, and you did mention that you found the speaker lacked spaciousness, so perhaps this is all related.