El único problema es que no funciona en modo privado.
Una sugerencia sería un botón de "Aceptar" al lado del cuadro de texto.

It works and I can finally access a specific website in the language I require. I would like an interface which lets me set different locales for different websites.

However, it is sufficient for my rare need to access this website and it helped me a lot. Thank you.

Works as advertised in Firefox 53.0 after the author pushed a bug fix. See https://github.com/callahad/quick-accept-language-switcher/issues/4.

Presentation for handling things like "q=0" (disallowed languages), multiple accepted languages, and invalid header strings could be improved, but those issues are merely cosmetic.

As already mentioned dropdown of values history will be great. Also it would be great to set specific values for specific domains.