I would suggest that you look at the context of this section a bit more.
This section states that Congress shall have the power to declare war. But it says nothing about the ability to make war. You seem to be using them synonymously.
The President actually has unlimited war making ability. This is his perview as Commander-in-Chief. The Congress, except for certain provisions accounted for in the Constitution, has no power over the military. For instance, according to your theory, if the U.S. were invaded, or if the Soviet Union had launched a nuclear attack upon the U.S., the President would be unable to respond until Congress had declared war. Well, what if the time frame alloted for a Congressional response was insufficient to the threat, or the theat actually did not allow Congress to assemble. Now, you might say that the Constitution makes allowances for such exigencies, but then I would ask you where does it do so.
If you look at the Constitution as a whole, or even simply the whole of Article 1, Section 8, this is one of the very reason that the Continental Congress limited the ability to budget for the army to two years. This is also one of the reasons why the House, as being closer to the will of the people then the Senate or President, was given the sole ability to generate expenditure bills. If the House, by virtue of the will of the people, does not approve of the President's use of the military, then it could eliminate the budget for the military, even to the point of simply not providing a bill for their funding. They could literally do nothing, then the army would go away. They could even make a budget law stating that the military could not utilize any money for a given action, such as attack Libya.
The fact of the matter is, Joe Biden is wrong.