This is quite good, and it briefly hits a couple of the more obvious ways in which naively applying an FFT will lie to you (superposition, harmonics).
If you really want to kill it, you could always come in with a learning model which accounts for the characteristics of the instruments and the way in which the sound evolves over time. An FFT is effectively looking for the integrated power in a frequency band. This is often more than enough to do the job, but it misses potentially useful information found in the time domain.
