It sort of worked during 1944. But the D-Day drop of three airborne divisions was a disaster, at least from a planners point of view. The night drop was disrupted by poor visibility (which, of course, was expected) and enemy anti-aircraft fire (also expected.) But since you can't really practice this sort of thing (can't simulate anti-aircraft fire), you learn by doing. What was learned during World War II was that large scale airborne drops, especially at night, put the troops in unexpected places. Trained to form into their units after a drop, troops dropped at night found themselves scattered all over the place. But paratroopers are trained to use initiative, and this they did. Seeking out and fighting whatever enemy troops they could find, the paratroopers caused chaos in the enemy rear area. But once daylight arrived, the enemy got the upper hand. Without the cover of darkness, it was realized how few, dispersed and lightly armed the airborne troops were. At that point, either friendly ground forces showed up, or the paratroopers began to get chopped up. This was the experience in every major airborne operation.
After jumping onto the island of Crete in 1941, the German parachute division got beat up pretty bad, and without the timely arrival of airlanded troops (via a captured airfield) the Germans would have lost. As it was, the German paratroopers took such a beating that they never again attempted a major airborne operation. Part of the German problem was two bad practices that American paratroopers noted, and fixed. First, the German parachutes were designed so that the paratrooper could not control exactly where he would land. Second, they jumped without their weapons, which were placed in containers that were dropped separately. American paratroopers were trained to work the risers (fabric connection the trooper with his parachute) so that a specific landing area could be selected. And American paratroopers jumped with their weapons, and were ready to fight as soon as they landed and unhitched their chutes.
Russia also formed large airborne units, and lost them whenever they dropped them. But it was noted that smaller airborne operations (from a few squads to a few companies) tended to be more successful. In particular, parachutes were useful for commandoes and spies.
After World War II, only the United States and the Soviet Union maintained large airborne forces. Many other nations had a battalion or a brigade of paratroopers, mainly as an elite intervention force. This, in fact, was what the American and Russian paratroopers were used for as well. As a consequence, they weren't used much at all. More often, there were smaller drops of elite troops. Sometimes they were paratroopers, at other times they were even more elite forces like American Rangers or Special forces, French Foreign Legion, British SAS or Russian Spetznaz.
The idea of mass parachute jumps still remain on the books. But as a practical matter, the airborne generals remember their history, and use smaller jumps, coming in via an airport or moving into battle on helicopters rather than everyone parachuting down.
Airborne troops are a 20th century invention that looked great on paper, was hugely popular with the troops and the public, and never really worked as intended. As a result, there hasn't been a major airborne drop since 1945 (and that one turned out to be unnecessary.) The basic concept is to drop large infantry units (from several battalions to several divisions) behind enemy lines and thus disrupt the enemy army.