> How the heck does a US president have military powers so powerful and broad? If congress can only declare war [...]

It's been this way since the Vietnam war, it not the Korean war. Every president since then has been able to engage in relatively small military operations without congressional approval. And the UN is what ended formal declarations of war, too. Basically Congress can stop military actions started by the President by taking the money away or not providing it to begin with, but if the operation is small then it's a fait accompli before Congress can do anything about it.

See the Bay of Pigs, Vietnam, Cambodia, Grenada, Iraq, Somalia, Serbia, Libya, Syria, etc.