It sounds sort of like you're talking about the effect of EQ on the whole summed signal. I'm I reading that correctly?
Well, that's what happens when you EQ the resulting stereo recording. You adjust everything together, all by the same amount. In a polar sense, the difference in frequency and level differences with direction remain unchanged.
What I'm talking about is EQing the direct on-axis response differently from the collective off-axis response. In the case of a directional microphone, the difference in on-axis response and the overall integrated response from all directions is 'baked-in" by the manufacturer, often dictated by engineering and manufacturing cost issues.. In the case of my surround recordings, I can tweak the the forward, left, right, and backwards responses differently if necessary. The analogy breaks down in the sense that a single microphone response is mono, and what I'm doing is multichannel, but the overall effect of recording a sound in a room is the same. The Schoeps PolarFlex system mentioned previously is a more direct analogy. It allows one to modify the polar response of a single virtual microphone, and do so by frequency (across three different ranges if I recall correctly), although it does not allow for changing EQ by direction.. at least not directly.
I think the degree response absolutely changes on application, not necessarily pickup pattern as much. If I'm recording an acoustic guitar in isolation I want way more off axis response than if I'm recording the same instrument with the same mic in a live environment.
Not sure what you mean by "degree response" there. Sounds like you may be mixing up the reverberant sound of a room, with the off-axis behavior of a microphone. They are related, but not the same thing. You can most easily adjust the direct/reverberant pickup ratio (in other words "how much room you hear along with the direct sound from the guitar") by changing the distance of the microphone from the source, or by moving to a room with different reverberant qualities, or both. You can also adjust the direct/reverberant ratio by changing the polar pattern of the microphone, but to a much less greater extent, and within a far smaller possible range of adjustment.
When you say "If I'm recording an acoustic guitar in isolation I want way more off axis response than if I'm recording the same instrument with the same mic in a live environment." that over simplifies things too much, in several ways. If you are recording a guitar in isolation, you might be "isolating" it partly by using a tighter polar pattern, and/or by moving it away or blocking it from other sound sources, and/or by placing it in a room by itself. Either way, that says nothing about the direct/reverberant balance or how "live" or reverberant the sound is from the isolated guitar. Could be super dry, or super reverberant, but still isolated. The appropriate microphone polar response and mic'ing distance is going to depend on the sound you're after.
Likewise, recording an ensemble in a live environment, it all depends on multiple things: the sound of the room, the distance of the ensemble from the microphones, the type of music, the presence of a PA or not, the intended use of that particular microphone channel or pair, and what type of sound you are trying to achieve in the recording. Since all polar patterns can be and are used for recording in "live environments", there is no hard fast rule as to which will alway be most appropriate.