There are several causes for this and as one might assume, they are of economical rather than military persuasion. I decided to break them down into three main areas that I'll try to address below.

Excessive cost

First, the possibly most obvious reason was the economic burden of such formation. Hussars required high-quality armament and mounts, making the procurement and maintenance of such troops very cumbersome for the treasury. Hussar's armour were usually both lighter and more durable than typical heavy armour of 17th-century curaissiers, more often than not made from the the best material available. Another expensive element was the hussar's lance, that had to made locally as this specific weapon had no counterpart anywhere else. Last but not least, each hussar companion (i.e. nobleman obliged to provide all required arms and equipment for himself and at least several retainers, generally a person we think about when using the term 'Polish hussar' or 'Winged Lancer') had to provide several combat horses, to accomodate for potential combat losses and accident. According to the ledgers and memoirs of the Polish officers and noblemen from the mid-to-late 17th century, the cost of a single well-trained combat horse was 3-4 kilograms of silver, what was 4-5 times more than the average cost of personal equipment (weapons and armour). An average cost of deploying a hussar team (a single lancer with several retainers) in a campaign was estimated in 1685 to about 5000 Polish zloty, roughly the cost of an average village (as evidenced by the purchase of two villages for 10.000 zloty by Jan Chryzostom Pasek in 1676). Duchy of Warsaw was a vestigial state, a constituent of the Rhein Confederation that in turn was a protectorate of the French Empire and thus could hardly bear the costs of raising a substantial hussar force even if it was possible, especially due to English blockade, heavily stunting the export of agricultural products (predominantly grain and hemp) that was a major source of income to the state treasury.

Lack of continuity

Second, this problem is not unlike the lack of archer units in 18th and 19th century that was caused not only by the low efficiently of the bow in comparison with modernised firearms and significant troubles with mass production of arrows but also the simple fact that there were virtually no people able to produce decent bows and teach both archers and their officers how to use such weapon in the field. Hussars used unique tactics that merged medieval idea of frontal charge with very precise unit maneuvers, making it possible to make several engagements followed by quick retreat if they were not enough to completely break the defender. To achieve this, hussars were trained to move in loose formation that was converging into tight one in the last stage of the charge, often in gallop. They were also able to perform a controlled halt and retreat in formation. All these maneuvers were facilitated by specially designated members of the formation (usually the most capable veteran cavalrymen) who, as one can imagine, required a substantial skill and many years of practice.

Such training required a steady traditions and constant transfer of skills actively used on the battlefield. This became more and more difficult in the early 18th century, when the numbers and quality of the heavy cavalry in Polish-Lithuanian Commonwealth declined due to financial and social reasons as nobility slowly withdrew from its direct involvement in warfare and preferring political or economical careers). Historians agree that hussars were officially deployed for the last time in the battle of Kliszow in 1702 (as a part of joint forces of Poland and Saxony). Eventually, since mid-18th century, hussars became purely ceremonial and representative troops with no military value. In 1775, polish Parliament decreed the complete dissolution of the hussar and armored cavalry that were to be substituted by the National Cavalry modeled after its Western counterpart. On a side note, this reform completely resigned from the use of protective equipment, and so Polish cavalry, using what was essentially medieval formation until early 18th century, was since 1775 composed only of light cavalry (with an exception for a short-lived 14th Cuirassier Regiment, disbanded after suffering heavy losses in 1813), even though many other countries (e.g. France) retained the units of armored cuirassiers until early 20th century. Thus, when the Duchy of Warsaw was proclaimed in 1807, there were basically no people capable of training hussar formations and definitely no people who ever witnessed combat use of such troops. Now, it is not to say that such formation could have not been formed again given adequate time, research and financial support but as the Duchy of Warsaw existed only for 8 years, during which time it was occupied for almost 2,5 years by Russian forces, there was simply no time to perform such a reform.

Unfamiliarity

Third, the changes in the tactics and technology itself made the heavy cavalry less and less useful on the battlefield. We need to remember that significant factor of the army's value is its efficiency and versatility, so introduction to foreign elements that are unfamiliar to officers usually does more harm than good, especially on the logistics level. Although heavy cavalry retained some effectiveness for sudden flank attacks, the development of the modern firearms, proliferation of field artillery and use of tiraillere made the use of heavy cavalry less efficient than before, and charges were rarely really effective unless performed against already shaken opponent. Thus, the formation of extremely expensive unit in place of significant boost to infantry or artillery in the era when heavy cavalry was considered as auxiliary formation at best, was not something many people would treat seriously in the early 1800s. Especially given that Duchy of Warsaw was a protectorate of France and Napoleon seemed to focus his tactics on infantry and artillery, so he most likely would consider resurrection of winged hussars as a nonsensical loss of money, given that he would be unable to use such troops efficiently on the strategic level because of lack of experience with such type of cavalry.