ms, the thing is that when a transformer saturates (either because it is "too small" or the signals are too large for it--you can look at the problem either way) (and "too small" is a simplification, as you know--but as a first approximation it'll have to do for now), let's not confuse cause with effect.
Yes, saturation is far more likely to occur with strong low-frequency signals--the distortion curves bend radically upward at the low end, which is part of why all "vintage" microphones (back when the inputs of all studio mixing desks always had input transformers) had filters at 40 Hz or so, some sharper than others.
But when saturation takes hold, the effect is not that the low frequencies are attenuated--instead, the entire signal, from top to bottom, becomes wildly distorted. The waveform all becomes One Big Ball o' Fuzz™. An apt term that people used to use was that the transformer "blocks" the signal when it saturates. It's like being choked.
If you send single-frequency sine waves (yes, I know that's redundant, and also says the same thing twice) through a transformer that's too "small," at very low frequencies you'll find a narrow range of levels at which the output is a distorted copy of the input, where the distortion is still listenable, and possibly even attractive for some purposes. But that's only with single-frequency signals, and only within a very narrow range of pitch or dynamics. If you go lower in frequency (= pitch) or higher in level, or add other tones, the amount of distortion increases very rapidly and the character changes drastically. Soon it's no longer musically interesting, except perhaps as a symbol of chaos and destruction. Which can be cool, too, except that it all tends to sound more or less alike.
A lot of the misunderstandings that people seem to have about tubes and transformers in audio come from the experience of guitar and bass players with tube amps. The presumption that transformer distortion will warm up the low-frequency character of a full-spectrum signal (as contrasted with the mostly single notes that bass players usually play) seesm to be one of those misunderstandings.
--best regards