I'd be very interested in what the true experts on this subject have to say on the matter, but my impression from reading and observations from collecting lead me to believe that Birmingham was the primary source of innovation and expertise, while London's success was largely based on the acquisition of Birmingham's techniques, craftsmen/artists and products, and then greatly enhanced by geography and their own brilliant marketing skills. (Note that many sharp Birmingham firms opened London shops to sell their wares.)
Most would agree that Birmingham could build "best" guns every bit as good as anything London built, but didn't make it their trademark the way London did. I generalize, of course, because both could boast the talent to create whatever their clients sought, but the Birmingham trade made its money supplying the guns (sporting and military) that built an empire, while the London trade made its money supplying guns to the elite who ran and lived off the empire.
It seems to me that the arguments in this ancient debate revolve around a difference of opinion as to the definition of the classic, quintessential British sporting gun ... regardless, it was a tool that 99 percent of the world couldn't afford, but was it a working man's Westley Richards or Greener boxlock, or a peer's exquisite and super-expensive Boss or Purdey sidelock? TT