About cardioid vs. omni w/r/t bass response: When a condenser microphone is constructed as a "pressure transducer" then it can have essentially perfect response down to any frequency that you might choose. You could even go all the way to DC, but it wouldn't be wise, because such a microphone couldn't adapt to changes in barometric pressure and (for example) its capsule might burst if it were carried or shipped on an airplane. As a result all microphones that are built on this principle have a safety opening which limits their acoustical response to some frequency such as 1 or 2 Hz. Generally the circuitry of the microphone imposes a further low-frequency rolloff of some kind, which varies according to the intended application. Note that not all omnis are pressure transducers. Multi-pattern microphones that have an "omni" switch setting almost always contain a pair of back-to-back cardioids in one capsule head, and if so, their characteristics are generally somewhat different.
A low-frequency rolloff and/or other response reduction might be applied by design to the built-in microphones of a recorder even when it has XLR or other inputs which have flat response down to, say, 20 Hz. If directional microphones are built into a recorder, they will be sensitive to wind and solid-borne sound (e.g. handling noise, or noise conducted through the desk or table that the recorder is resting on), both of which occur mainly at low frequencies. No one would knowingly choose to record a serious music production via the built-in microphones of a typical hand-held recorder; it's widely understood that external microphones must be used for the highest quality sound. So we could both "be right" in that a high-quality recorder, designed to work well for serious music recording through its microphone or line inputs, might still have limited low-frequency response when the built-in mikes are used. A low-frequency limit makes sense as a "safety" precaution, while an overall reduction (for several octaves above the cutoff) of the low frequencies greatly improves the clarity of speech in reverberant and/or noisy environments.
The attached filter curves are those of the Nagra IV-S recorder, designed ca. 50 years ago by now. They are very well chosen from experience in dialogue recording for film. "S+LFA" is the curve that's used when recording speech in a noisy and/or reverberant space; it makes an enormous improvement in speech intelligibility without sounding "thin".
Finally, if you take a pair of cardioids that have good low-frequency response and a pair of pressure transducers with identical measured (on-axis) low-frequency response down to some frequency such as 40 or 50 Hz, and you make stereo recordings with both pairs in a normally reverberant space, the "feel" of the results--the sense of spaciousness--will still be very different at the low end. Partly this is because you would almost certainly space the omnis apart some distance and/or put an obstructing device between them (at least for the high or mid-high frequencies), whereas you might keep the cardioids close together or even "coincident" (a technique that IMO should be used only when a very high degree of mono compatibility is required).
But partly this is also because of the way that any directional pattern interacts with the "room modes" (low-frequency resonances produced by the room's rigid, physical boundaries); this even occurs in mono recordings and the effect was well known before the stereo era began. High-quality music recording in (say) the 1940s and early-to-mid 50s was generally done using a pressure transducer for the main microphone as much as possible, though good room acoustics are required for that to work well. Thus record companies were very choosy about where they would record their most important productions and the way in which the sound sources (instruments, voices) were physically arranged on the stage or floor. Directional microphones were regarded as "problem solvers" for dealing with inferior acoustics, problems of interference due to unwanted sounds, and/or special problems in balancing an ensemble.