Louis XV did wait until his majority, he actually waited even longer before taking power.

Of the two important periods of regency during the modern era, Louis XV's is probably the least adapted to your question because of his character. Indeed, with the king turning 13 years old in 1723 the regency was legally and officially over. However, Louis showed no interest whatsoever in his public and political duties. He reigned but did not rule. Philippe d'Orléans pursued the policies he started as Regent as principal/first minister until his death several months later in December 1723. Cardinal Dubois who played an important role alongside the Regent then took his place until 1726 when Cardinal Fleury, former tutor of the King, became the strong man in the kingdom. This lasted until 1743 and only then did Louis XV really took power and continued, or at least tried, on his predecessor's footsteps (without a principal/first minister). Now could he have broken the regency ? That we cannot know. But we have the example of Louis XIII, whom became an adult in 1614 and should have started to exercise power.

A regency is always a critical moment for the political development of a country. Basically it means that the previous monarch died too early or is incapacitated for some reason. A such, political factions surrounding the royal power can rightfully see an opening to press their influence on the regent and/or a young monarch. This actually happened soon after Louis XIV's death in 1715 as his illegitimate son, the duke of Maine, was supposed to become regent but Philippe d'Orléans disregarded Louis XIV's will and became regent with the support of the Parliament of Paris (which I will talk about later). A century before, Louis XIII, suffered from the great influence of his mother Marie of Médicis and her favourites. The regency should have been over by 1614 when the king turned 13 years old but, de facto, the Queen Mother remained in power until 1617 despite the late Henri IV's wishes for a smooth transition of power in case he died too early. Sure of her son's weak spirit and will, she and her favourite Concino Concini were basically (and to keep it short) detaining most if not all the power. Concini increased is own wealth, disgraced courtiers and named the ministers, including Richelieu whom he introduced to the government.

But young Louis XIII was not the puppet many thought he was. Not to mention that Marie, Concini and those close to them were hated by the court, the aristocracy in general and people of Paris. They were hated because they were Italians, hated by the Protestant faction because they were Catholics and hated because it was becoming very obvious that they were monopolizing the power. In 1617, Louis had a circle of close friends and supporters at court which were all eager to see the Italians out of the way, including the Queen Mother still exercising a regency that was lasting too long. And the king was eager to put an end to this and to start ruling by himself. One of Louis' guards shot dead Concino Concini, in public when he entered the castle of the Louvres and his faction was disbanded.

As you can see, it took a lot of efforts for Louis to ensure that he could put an end to the regency that kept governing France even after it should have been over. He had to gather and be sure to have enough support at court, to have a faction of people close enough that they would accept what looked like a coup.

Another point that need to be pointed out is the role of the parliament of Paris. Its role is often overlooked because popularly early modern France is imagined as an absolute monarchy where the monarch can do whatever he likes, which is not true, especially during the 18th century. The parliament of Paris had, among its prerogatives, the right to control that the King acted according the laws. France had not written constitutions during the Ancien Regime but there were a set of laws known as the lois fondamentales du Royaume (the fundamental laws of the Kingdom), built over the centuries by tradition, which the monarch was bound to respect. And he swore to do so during the coronation ceremony. According to these laws in case of minority, exile or any other case where the monarch cannot fulfill his duty, the power was to be in the hand of a regency. In reality, no strict law ever existed to define what a regency was except that there was to be one if the monarch was in one of the above conditions. So, even if predisposed before his 13th birthday, a king could not simply take over a regency, risking a conflict, a fronde with the parliament. And a conflict with the parliament was always a very dangerous thing to do, as Louis XV would learn later during his reign. Note that if Louis XIV muzzled the parliament of Paris, partly as a retaliation for the civil war caused by the conflict between royal and parliament powers (the Fronde, 1648-1653), some of the powers of the parliament (mostly the right of protest, droit de remontrance) were restored during the regency of Philippe d'Orléans in 1715 as part of an agreement with the members of parliament in exchange for their support in discarding the duke of Maine from the regency.

In the case of Louis XV, a psychological approach can also be done. He was a very shy child, not very strong in terms of health and could simply not act against those in powers, mostly against the Regent who was after all, his closest blood relative. Also, but we cannot say whether or not Louis XV was aware of this, the regency was doing rather well. Except for the fact that Orléans disregarded the will of Louis XIV and made himself regent, he never attempted anything that could harm the young king or go against the royal power. It could have been tempting honestly because he was, even if not the Dauphin, the closest male relative of the king. When the regency was over it was only because Louis XV did not personally assume power that he became principal minister. I should point out that a position similar to the British Prime Minister never existed during the Ancien Regime and politicians such as Mazarin, Richelieu, Orléans and Fleury were only de facto principal/first/prime ministers but it never was official.

A good biography covering Louis XV's childhood and the regency can give you a very good picture of why Louis never attempted anything. I would go as far as to suggest that it never crossed his mind since he refused to assume power and basically extended the regency for another twenty years. Louis XIII's case also shows that it is not an easy thing to do to go against those exercising the real power.

I do not know if there are many biographies of Louis XV in English but one should be enlightening to see how his childhood was.

Louis XV by François Bluche

La Monarchie des Lumières, 1715-1786 by André Syzberg