Of the ESI spectra that I have encountered, positive & negative mode, the detected species always have charge (positive or negative) corresponding to the charge mode that the machine is in. Is it therefore the case that the ESI detector can only detect charged species, or that the overwhelming majority of species are charged?
I feel you have a misconception. ESI is an ionization technique not a detection technique. The sole purpose of the electrospray ionization is to impart a charge on a molecule in the gas phase. The molecule could be neutral, anionic or cationic in the solution phase. The inventor of ESI used to say that ESI makes the elephants fly, i.e., very large molecules could also be ionized in the gas phase.
The important point that you must learn is that if you have an ionized molecule in the solution, it is not necessary that when you bring the molecule in the gas phase, it will have a electrostatic charge. It may become a neutral molecule.
In order to "sort" molecules according to their mass to charge ratios in the gas phase, it is best to ionize them, i.e., remove an electron or add a proton so that the molecule of interest acquires an electrostatic charge in the gas phase. Note that I am emphasizing gas phase again and again. Forget what is the state in the solution. Almost 100 years ago, physicists had worked out how "charges" will move in an electric or magnetic field predictably. If the molecule is electrically neutral it will not interact effectively either with electric or magnetic field. Hence you would not be able to sort them according to their mass/charge rations. This is the job of the mass spectrometer.
Once your molecule has a charge, it is very easy to direct them here and there in the mass spectrometer and even focus all the ions at place (one of the major advantage of ionized molecules), before exposing them electric fields or magnetic fields.