When I crossed the border on foot there were no officials there at all. This was before I became British so I sought out a Spanish official later, who wasn't very interested in my passport either.
For a US citizen I would recommend the original plan as in the OP. Or, fly into Spain, go to Gibraltar and back, then fly out of Spain. Otherwise you will have a Gibraltar entry stamp but no Schengen entry stamp; or a Schengen entry stamp and no Schengen exit stamp. This probably won't cause any problems, but it could.
Alternatively fly in and out of GIB, but cross the border to stay overnight in a cheaper hotel in La Linea, then (barely) cross the border the next day for your flight. I stayed in a hotel which was 5 minutes walk from the border - I can't remember what it was called but it may have been the Ohtels Campo de Gibraltar based on a Google search; the hotel I stayed in was fine, similar to an older UK Hilton (if you know what I mean)