The thing about transformer saturation is, while it is indeed triggered much more easily by low-frequency signals, the distortion then affects everything passing through the transformer at the same time--all frequencies, not just whatever's loudest. Also, the amount of distortion rises VERY steeply as the signal level increases, and quickly becomes a near-brick-wall situation. You only get a narrow dB range in which the effect is clearly audible, yet doesn't make everything sound like garbage.
I can understand using transformer saturation as an effect if you're, say, an electric bass player and you want to make your sound more complex and interesting. It could work then, because it's just the one instrument, playing generally just one note at a time, at fairly well-controlled signal levels. The transformer and/or tube circuitry can then be doing its thing on the bass part alone, while the drums and vocals and guitars and keyboards are independent--neither causing the bass to distort more, nor being affected when the bass does distort. The bass player rehearses, sets up and tests, and sets levels, so that just the right signal levels will pass through the transformer during the performance--not too low, not too high, but just right--and that's really what makes this approach capable of working repeatably.
But if you're recording the band, then you have to consider that (a) a saturating transformer in an acoustical recording setup distorts ALL the sound that's being picked up, that (b) you can't usually predict the exact levels that you'll get during a concert, and that (c) you can't usually monitor moderate amounts of distortion in a recording reliably during a concert, nor adjust the signal levels so that what's going through the transformer is neither too low nor too high, but just right.
A transformer isn't a magic ingredient, in other words. If it isn't needed for connecting specific pieces of circuitry together, it's almost always better not to use one at all. The idea of throwing it into the "recording chain" as a kind of flavoring is fantasy-based engineering. Since you can't predict what effect it's going to have on the recording, it makes far more sense to record cleanly and then experiment with various alterations of the sound in post-production, if you really think that there's a chance you can improve things. That way you can compare the actual recording with the tricked-up version and see which one you prefer, and you can even have it both ways if you like. But if you put a distorting device in the signal path during a live recording, you can never take back what it is doing to the sound.
--best regards