String growl, in its essential form, is the string vibrating in its overtones, also called partials or harmonics, so it has a more complex character and wave form. That's why rounds, especially stainless steel rounds, are usually considered more growly, since they are more flexible and therefore vibrate in their overtones more freely. This manifests itself as more mids in the overall tone. But it's not just raising the mids on the amp. It's the overall wave form. The Fender Flatwound strings were changed in their formulation a few years ago. The new ones are more flexible, so they vibrate in overtones more freely, and therefore have growl.
The old Fender flatwounds were a less flexible string, and did not vibrate in their overtones as much, so they did not have growl. They were more mellow in tone. Many people miss that tone. It is the tone of many famous recordings of the last sixty years, and is a tone that supports a band well, ancient or modern.
The flatwound manufacturers have made great strides in their manufacturing techniques. For example, the ribbon on a new Fender flat E string, whether 100 or 105, is narrower than the A string ribbon wrap, and is slightly beveled. This contributes to their rough texture, more flexible feel, and their more growly nature, compared to the older formulation.
D'Addario, on the other hand, has a very tight wrap with more chromium in the mix, so they are more shiny, the string is less flexible, and when it loses its new string zing, is very mellow, almost thumpy tone because it is so tight. The upside of this is finger gunk doesn't get into the core of the string, so it lasts a long, long time.
Ernie Ball flatwound string construction is in the middle, so yes, the tone is in the middle as well. If I were to go back to Chromes for any reason, I would use an Ernie Ball E strings to keep a more consistent tonal spectrum. Indeed, when my set of new Fender flats were starting to mellow out, and my E string started going darker than I wanted to play, I also purchased an Ernie Ball 105 to keep tonal consistency. It's just a great flatwound E string: defined, but not overbearing or thumpy. Probably one of the most underrated flatwound strings out there.
GHS seems to have the more rectangular ribbon wrap, but if anything, they are wrapped "looser" than some of the other strings. Ironically, this produces a "hollow" or "woody" tone that many people like, especially on a P-bass. I can only surmise that it has something to do with the core geometry and wrap ratio.
I do not use the TI or Optima (successor to the Maxima) flats because my hand perspiration eats nickel and their pure nickel wrap turns my hands black. I don't want to pay the price for Pyramids, and there are several boutique manufacturers which I have not tried, and probably won't, unless I find a good set used.