For whatever reason most Americans use the word "customs" when they are, in fact, referring to immigration, when traveling internationally.
Immigration - which is the process of becoming a US permanent resident and/or citizen - is handled (mostly) by USCIS.
Other visas are handled by the State Department (foreign ministry).
Not an expert, but this is my understanding.
I have Global Entry but I don't think the US even has a customs form any longer.
>very few exceptions
You've got it backwards. If you're an American you're probably traveling through freeports or low tax regimes like Singapore, UK, etc. and don't realise how regressive most regimes are. In places like Hungary, Angola, SEA -- where tax can be in the range of 30-50% you will be lucky not to be shaken down by a customs agent before leaving the luggage carousel.