In AAS, the flame doesn't really excite atoms so much as blast the sample apart (atomize) so that atomic spectral lines can be observed. In condensed phase, atomic absorption lines are usually "washed out" by the various molecular vibrations and so forth. When the sample is atomized, atoms become well separated in the gas or plasma state, where their absorptions can be easily observed. That's a pretty big simplification but maybe you get the idea. (There is likely atomic excitation as well, with absorption originating from excited atoms. Starting with the flame temperature you could probably determine this with the partition function approach. I'm not really sure what percentage of atomic absorption lines in the AAS experiment arise from ground-state atoms and excited-state atoms. This will probably depend on the element.)
Compare this to ICP-AES, which uses an RF coil (large electric field) with temperatures four to five thousand degrees hotter. In this case, samples are not only atomized but electronically excited (and ionized) as well, which then decay back to their respective ground electronic states by emission (i.e., fluorescence).