The Korean War, the Vietnam War, Operation Desert Storm, the Afghanistan War of 2001 and the Iraq War - none of them were initiated with an official vote by congress.
This is inaccurate.
President Truman did break from precedent when he initiated a "police action" (read: not a war) on the Korean peninsula in mid-1950, and perhaps took advantage of an impending July recess when that "police action" started to look more and more like a "war". There were however plenty of votes that Congress held during the remainder of the year that supported the actions that President Truman was taking. There were quite a few people serving in congress that raised a fuss about it at the time, but ultimately went along and funded the operations.
Congress passed H.J. RES 1145, dated August 7th, 1964 after a US ship was attacked in international waters (the Gulf of Tonkin, which is why this legislation is more informally known as the "Gulf of Tonkin resolution"). This gave President Johnson authority to increase U.S. involvement in Vietnam. After awhile, Congress decided to retake some control back from President Nixon when they passed the "War Powers Resolution" in 1973. From their point of view this gave the Commander-in-Chief the power to protect American interests while at the same time limiting the scope of any action possible by placing a time limit on what the President can do unilaterally. A President can commit forces if they feel it necessary, but must notify Congress within 48 hours of doing so and only allows action to proceed for a total maximum of 90 days. After that, for any military involvement by the United States to continue, Congress must authorize it.
Because of that legislation, Congress now passes what people call "AUMFs" (authorization for the use of military force). Desert Storm was authorized by a bill titled "Joint Resolution to authorize the use of United States Armed Forces pursuant to United Nations Security Council Resolution 678". Military operations in Afghanistan were also approved in a similar way with a bill titled "Joint Resolution to authorize the use of United States Armed Forces against those responsible for the recent attacks launched against the United States" which was signed into law by President Bush on September 18th, 2001 (one week after 9/11).
All of this is to say that Congress has, in effect, ceded some of the power granted to them under the Constitution to the Office of the President, but not without reason. The ultimate answer to your title question, however, is that we have absolutely no idea. The only people with standing that can challenge in a court of law any given action would be Congress itself, and it has never chosen to do so. And, really, why would it when it can just pass legislation targeting anything the President does that they don't at least tacitly agree to? That way they can get what they want without having to worry how a squeamish Supreme Court may rule. It doesn't help matters that pretty much every single President since Nixon has in some form argued that the War Powers Act itself is unconstitutional. Additionally, there's not a single member of any branch of the Armed Forces which would listen to orders given to them by any member of Congress which contradict orders given to them by the President because doing so is likely to land them in jail. What we've more or less settled in to is that yes, Congress controls the power of "war making" by controlling the funding of the operations themselves, and the President can fulfill their obligation to protect and defend the United States and it's interests regardless of whether Congress is currently back home kissing babies at the moment.
Below is a non-exhaustive list of pre-World War II military actions that the US participated in or against foreign territories without a formal declaration of war:
*- The Indian Wars is actually a 300+ year long intermittent conflict between European Settlers (after awhile referred to as "Americans") and different groups of Native Americans who had lived on the continent for thousands of years. This one bullet point could easily be deconstructed into dozens of individual conflicts.
