The USA does not recognize citizenship other than US. If you are a US citizen you are supposed to enter the US with your American passport, so the INS may not accept his Mexican passport as ID and might give him a hard time. I wouldn't recommend that he do so until he has his US passport. It is always best to avoid conflict with the INS (or whatever they're called now).
I'm a dual citizen myself, so I've dealt with this all my life. It is true that the US used to not recognize dual citizenship, but that has changed. You are even allowed to serve in foreign armed forces if you are required to (like in Israel or Switzerland), although you might have issues if you become an NCO or officer, as that would imply that you have gone above and beyond what is simply a required period of service.
When I fly to and from the US and Switzerland (my other country of citizenship), I show my US passport at US immigration and my Swiss passport at the other side.
In the OP's particular case, you don't have a US passport yet, so you don't have much choice. I would simply travel on your Mexican passport, but keep your US Citizenship correspondence with you, so that if they do challenge you at immigration, you can show them the paperwork and explain that you are still waiting for your new passport.
But again understand that you are violating US law and you can be fined. It's like speeding, you may not get caught but that doesn't make it legal and if caught you have to pay the fine.
Just say you didn't do it on purpose. Seemed to work for Hillary...
Democrats haven't been this angry since we took away their slaves.