AES is not the same as AAS. You seem to be getting them two confused.
AES is superior to flame tests in every way. Flame tests only give you a macroscopic "average" view of the emission spectra, and plus the flame may be too weak - even if it is strong enough, the emitted wavelengths will be invisible to the naked eye (UV - the high energy wavelengths of light).
For AAS, the flame in the atomiser is supposed to bring the atoms to their ground state (not 100% sure how, but you're supposed to assume this happens). I haven't seen an official explanation against why the emission caused by the flame does not affect the final results in the AAS, but I suspect this is due to the monochromator placed after the absorption, which selects a particular wavelength of light (the one that is being absorbed), ignoring the other emitted wavelengths (I'm aware this isn't a complete explanation). It's a very good question that my friend asked TSFX, who didn't have an answer for it. You probably don't have to worry about it.