Measuring the resonant frequency of an RLC circuit
The problem is, I feel, in your thinking. The unit impulse does contain all frequencies and therefore, the RMS value of the particular frequency that corresponds with your filter's resonant frequency is infinitely small. 

But, you might say that your filter's bandwidth is still quite wide so the energy of the spectrum around your resonance is not infinitely small. However, all the "in-band" frequencies that are "stimulating" your filter are incoherent and you can't expect to see those energies translated to one clear and obvious sinewave.