Would airlines still be profitable if you stripped away fees?

Yes, but hardly.

The nine largest US airlines will collectively report $2.4 billion in quarterly profits on $33.3 billion in revenues next week, according to Robert Herbst of the site

By my calculations, those airlines will have collected $2.25 billion in ancillary fees during the same period. If those fees are all profit (and there’s an argument to be made that a lot of it is) then the industry would have made only about $150 million in the third quarter.
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