I'm not sure what you mean by "on a small window such that this isn't really applicable"; can you give an example? As long as you accept the inherent time-frequency resolution trade-offs, there's no obstacle to using FFT on a small window. It's called the short-time Fourier transform (STFT), and it's used everywhere; it's probably used more than analyzing an entire signal, since we frequently want to know how power and phase change over time in a signal, and a full-signal FT can't tell you that.
FFT is very useful on a small window, but algorithms that improve the asymptotic efficiency are unlikely to be useful at that scale. With n=100 the asymptote doesn't matter. We'll still be using the FFT forever, I'm just skeptical that the frontiers of technology will be advanced by a faster FFT, since it seems like the coolest stuff is happening elsewhere. It's increasingly becoming the quick-and-dirty counterpart to the sophisticated-but-slow methods.
This is a useless post except that I wanted to highlight that wonderful little bon mot. Well done.
Computational resources is always the bottleneck in bioinformatics, quantum chemistry, or any sort of high data volume analysis or simulation, and the FFT is a fundamental and commonly used transform in all fields.
At least for people who still use computers to, well, compute things, a faster FFT is a huge deal.