What is the advantage of measuring IQ data non-digitally?
IQ demodulation is normally done with the "I" component being sampled at a rate that is at or near (including below) the carrier frequency of the signal, or the center of the band of interest.  Since this is well below the sample rate required to meet the Nyquist criterion (e.g. sampling at a rate greater than 2X the highest frequency of the signal spectrum), using the "I" component samples alone at that 1X  rate won't work, as you will get aliasing (likely scrambling together both positive and negative frequency spectrum above and below the sample rate into the one channel of strictly real I.F. or baseband data samples).  This will alias (make a mess of) your spectrum before you can even try to do any sort of Hilbert transform filtering.  (Unless you have a really steep filter before sampling that does the split into just USB or LSB for you.)  Sampling at exactly 2X the carrier has a similar problem.  Using the actual sampled Q component data adds information that properly keeps the positive and negative spectrum (the spectrum both above and below your "I" component sample clock) separated in the IQ I.F. or baseband samples.  

IQ sampling can be done using a sample clock (oscillator) that isn't at a frequency much higher (e.g. over 2X higher) than the frequency band of interest.  Instead, only a 90 degree phase delay is required at that much lower sample clock rate (Possibly also simplifying the required filtering as well as the sample clock generation.)