Place of birth is one of the data points which US CBP and many countries use Thus, it is potentially problematic and something I would want corrected.
Your profile does not show your location, but if you are near a Passport Agency, this is something you can have corrected in person in a few hours if you show up with the erroneous passport, the properly completed form and photo. There is a number listed for you to call on the State Department website, but you can also simply show up early (before opening time) and explain your predicament.
Do not make this about fault because that is irrelevant.