Accuracy of Goertzel algorithm for measure harmonic
The most common Goertzel algorithms are identical to calculating the magnitude of 1 bin of an FFT/DFT using a rectangular window.  As such, there will be rectangular window artifacts (so called "leakage") for any spectral content, such as harmonics, that are not exactly periodic in the DFT length, or Goertzel length.

Thus the accuracy of using just 1 bin may not be more accurate if you increase the length from one where all the spectral content is periodic to one where it isn't.  With a complete FFT, one can somewhat mitigate this effect by using high quality interpolation in the complex domain before taking the magnitude.