GC/MS is certainly the most universal at least for the most volatile constituents.
With the electron ionization spectral data base you may identify most molecules.
LC/MS with electrospray ionization may work for less volatile compounds by is less efficient for non polar constituents. NMR is much less sensitive than MS, and spectra with several hundreds of compounds will be extremely difficult to analyze except for the most abundant species.
My first answer was about identification. Quantification of the hundreds of constituents of essential oils is difficult or impossible. Some target compounds can be quantified more or less exactly using calibration procedures , for example using internal standards.