A corollary, would faster hardware help NOAA & friends? Do they already have the best available? Or are the models not able to take advantage of extra computational power?
(Disclaimer: I don't work in operational modeling)
NOAA just upgraded last year to a system that hits 8.4 petaflops, which is about the same as the European system, and the Japanese and UK systems are fairly similar.
More computing power would absolutely help the NOAA models (primarily GFS) increase resolution, improve the data assimilation method that generates the initial conditions, and increase the number of ensembles run. The GFS lags the European model in all of these areas. It would especially help increase the number of ensembles, since that is an embarrassingly parallel problem. However, NOAA also needs more researchers and funding. For example, model configuration changes not only need to be developed, but also tested to ensure that there aren't unexpected regressions in forecast skill. And, for example, the Europeans have put a lot of research and development into their data assimilation method, and it's one of the reasons they tend to outperform other models.