I personally dont think that us Americans were sitting idly by while the Brits made the best, outdated firearms in the world. I doubt a Frenchman or a German would agree with Brits making the finest weapons available either. The Brits surely didnt think they had the finest weapons during the Boer War.
Nobody could touch American design in repeating arms, pistol making, accuracy, etc during the late 19th and early 20th century. American design, sporting arms and military, had no equal. Only until Peter Paul Mauser came along with his brilliant design was there a real challenge to American repeater design.
The Brits flourished and were exceptional at producing an outdated sporting arm, the side by side shotgun and double rifle for many years, and they did an outstanding job of it.
The original poster was addressing SxS quality. My remarks were addressed to that question. Also, I recognize that there were some other makers in other countries (the few exceptions) like Johann Springer who made absolute best SxS guns. The British, as you point out, failed to develop much at all in the way of repeating arms... their market did not accept them readily.
Americans were definitely better/best at repeating arms for sporting applications. The Brits were very good at rifles as well, but their rifle markets (Africa, India, Scotland) requirements were very different from what the American manufacturers were serving. As a result the Brits kept producing those 'outdated' rifles that served so perfectly the needs of sporting clients in places like India and Africa. Horses for courses.
