every speaker will have as many power responses as there are rooms.
I think there's something to be noted, in that sound power response and "sound that we hear from speakers in rooms" - are still not the same thing even though the latter is greatly infleunced by the former.
Sound power is an anechoic, imaginary, and maybe measured but mostly "calculated" concept. Think of it as the sound being radiated into all 360 spherical directions... averaged.
What we hear in room is first the direct response, then the reflected response and it doesn't all arrive at the same time, minus whatever was absorbed or reflected by every object or wall in the room.
So rooms have their own effect but that's a bit different from the speaker power response in terms of the crossover. The crossover does a few things to what the speaker is radiating..
1) It affects the lobing behavior of the speaker. Even a speaker that's flat on axis, will likely have various peaks and dips off axis from interference between drivers, because their center to center spacing is rarely close together enough. Coaxials are of course less guilty of this as they are a true point source. The lower the order of the crossover, the broader the frequency region of the lobing. Even 4th order crossovers look pretty bad as you move away from the design axis. 110db/oct or higher look better IMO as they operate over a very narrow frequency range. For these very high order filters to work you probably want linear phase active crossovers, and you probably want VERY transparent and SIMILAR drivers so that the "difference" between the drivers isn't revealed by the lack of overlap (bear in mind, 4th order is 24db/oct so 110db/oct is bordering on ridiculous). Here is one speaker that used this kind of crossover:
http://www.audioholics.com/reviews/speakers/satellite/nht-xd-loudspeaker-system-review
2) It affects the total radiated sound in the crossover region - odd order crossovers tend to have a smooth power response, but can have a a peaky summing between drivers somewhere off axis (assuming the on axis is made flat) because (-3db) + (-3db) in phase sums to (+3db) somewhere (rarely on-axis). Even order crossovers tend to have a power response dip (which arguably we don't notice as much especially with a higher order crossover) which means that the on and off axis response will be pretty smooth, but the tweeter, and midrange for example, are actually not radiation as much combined sound near the crossover point, as they are at other frequencies. This is because they're both (-6db) + (-6db) which is what's needed to sum to (0) on-axis. Mathematically, even coaxials have this problem. It's pick your poison. Have two drivers reproduce the same sound at the same, but reduced level so they add up flat at one point, or have two drivers reproduce the same sound at different levels but such that they add up flat overall with the caveat that there will be some peaking somewhere??
3) A crossover means that you're using multiple drivers each with their own directivity index. So the crossover should attempt to match the directivity indeces. The drivers themselves have a power response and it should be reasonably matched. This is why many higher end speakers like Genelec, Revel, JBL LSR, Pioneer/TAD, KEF all restrict their tweeter's off axis response - because the woofer's natural dimensions are restricting this.
So when we refer to power response, we're refering to what the speaker is radiating. What the room absorbs or doesn't, is definitely REAL - it's just out of the control of the speaker designer. The speaker designer's goal should be to put the final buyer in a position to succeed as widely as possibly. Ideally there's custom designed NEUTRAL rooms as well. But for non neutral rooms, speakers shouldn't compensate(beyond tweeter level/baffle step controls), and for non neutral speakers, rooms shouldn't compensate (that is.. you shouldn't have to cut off half of what your speaker is radiating for it to sound good)!
Based on power response being an in-room measurement, I'd say it would be a speaker optimized the direct and reflected energy. Behold! We have a well placed Phil or SS.
The soundscapes and philharmonics, while they have very wide dispersion, i think are still well away from being considered "omni". The measurements you see are what, out to 50 degrees? They're fantastic. But omni speakers look the same at 0 degrees that they do at 180 degrees off axis, and on paper at least, vertically too (though not in execution as that would be a tiny 1/2" full range ball!)
On paper, an omni speaker radiates the same sound in every direction (thus in phase unlike a dipole or an open back mid) - a theoretically flat power response. So most omni speakers are actually voiced with a tapering axial (any axis really) response. This tapers the power response to more closely mimic box speakers.
Toole's research shows that in most rooms, not a flat, but tapering power response was shown to sound realistic. Damned if I know why (probably a mix of the speakers the audio is mixed on, plus the way flat response + tapering power response simply sums up and because perhaps some directivity is desired)