The reality is that they have abdicated the constitutionally specified power to declare war on other nations for political reasons. Generally, they are too scared to take a stand, and would rather let the president make the decision. Then they are free to cheer or snipe as the popularity of the war waxes and wanes.
Although it is not specifically spelled out in the Constitution, common sense would dictate that a declaration of war would, at a minimum, somewhere contain the phrase "declaration of war" within it.
"Authorization of force" is typical weasly lawyer phrase, produced by the hundreds of typically weasly lawyers who have come to infest our congress.
I think there is a lot of leeway in allowing use of military force without a formal Declaration of War, as even our founders used force in at least three actions before the first true War.
But I do think that since the 50s the cowardice of Congress has led us to this point.
I wish there had been a Declaration of War against Iraq, we certainly had the Casus Belli all lined up for it.
But we didn’t, and it will go down as another military action without a Declaration of War.
You raise some good points here, and this is probably one reason why Congress doesn't declare war anymore, but it is far from the biggest reason.
The real reason Congress no longer declares war is that a war declaration would effectively force the U.S. government -- from a political standpoint as well as a military standpoint -- to wage a full-scale war against a foreign nation, obliterate its infrastructure, kill many people, etc. The U.S. hasn't had any intention of doing any such thing in decades. Nowadays, we fight colonial wars whose sole purpose is to topple one regime, support another governing faction, and spend many years in what often ends up being a useless "nation-building" campaign.
In this kind of scenario, destroying a foreign enemy's infrastructure and laying waste to large pieces of real estate is completely counterproductive.
So we don't fight wars anymore. We just pick sides in the internal affairs of foreign countries.