Bit lost here, to be honest - What exactly did the US win in these engagements?

You might make a case for increased control of and importance in the middle east for Iraq, sure - that declined somewhat after a forced fumbling backwards out of Afghanistan and tanked on contact with Iran, so if that was the take home stuffed panda prize to write home about it's been well and truly run over by a road train and left in a ditch.

Clearly big wins for the USofA PMCs, but that's a little more narrow than "America".