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Jesse! Great questions, and you’re right about the way cardioid and omni polar patterns pick up sound, with cardioid picking up sound from the front of the mic/capsule and omni picking up sound from the front, back and sides of the mic/capsule.
But … it’s important to understand that regardless the polar pattern, these mics output a mono signal. An omni mic WILL capture sound on all sides, giving you a non-directional representation of the room, for example; however, it will be a mono recording of the room … unless you use two or more mics and pan them anywhere other than center.
In the case of an omni mic when we listen to the recorded audio on a stereo playback system, the sound does come out of both speakers but equally so, thus it is mono. One mic – even when in omni, picking up sound from all sides – can never reproduce stereo sound. Gotta have 2 or more mics to recreate the stereo image you hear with your two ears 😉
Does that make sense? I’m realizing while typing just how odd and difficult these concepts are to write about. Much better to demonstrate with sound and/or video! 🤘