Through her position as court favourite Pompadour wielded considerable power and influence; she was elevated on 12 October 1752, to duchess and in 1756 to lady-in-waiting to the queen, the most noble rank possible for a woman at court. Pompadour effectively played the role of PRIME MINISTER, becoming responsible for appointing advancements, favors, and dismissals, and contributing in domestic and foreign politics.[20]
The royal consort (the Queen) was sent there by some other royal/noble family to gain influence in France (maybe I'm wrong?), so if a random woman that the King fell in love with, can just supplant the consort and many times, have even more influence than her, why would anyone want to send their daughter to the king?
In the first place, why did the king and the court at large, permit the mistress to have so much power? France was pretty sexist, even at the time, no? The Salic Law and whatnot. So why was a woman that was not even a royal consort, and (sometimes) not even of noble birth, allowed to act almost like a PM?
I'm so glad you asked this question!
In early modern France, the royal mistress was not just "a random woman that the King fell in love with" - maîtresse en titre was an official title at court. The first was Agnes Sorel (1422-1450), given that title by King Charles VII; unfortunately, we don't know exactly what prompted him to take the step of officially recognizing his mistress and making her essentially the first lady of his court in the face of ecclesiastic disapproval, but it was a decision the reverberated down the ages. Succeeding French kings would appoint their own maîtresses en titre, who would then be entitled to estates and incomes, would entertain and emotionally support their kings, and would lead intellectual, social, and cultural circles during the periods they were in power, giving patronage to writers and artists and being a path to the king's ear as well has having it herself. The kings in many cases slept with other women at the same time (see this previous answer of mine on Louis XV's house in Parc-aux-cerfs), but his official mistress fulfilled the role we would expect a wife to play. She filled an additional role to the public as a safe target for criticism, in comparison to the absolute monarch himself, and was traditionally reviled as a bad counselor when he made missteps or as a money-hungry vixen when the treasury was low.
French queens still had power. They still socialized at court and gave their own patronage, and they were also pathways to their husbands. Perhaps more importantly, they would bear children, one of whom would be the next king - and France had a history of young kings whose mothers acted as regents during their minority, as well as adult kings who were strongly influenced by their dowager-queen mothers. (For example, Catherine de' Medici and Anne of Austria.) Unlike the official mistresses, too, they couldn't be retired at the king's whim, except potentially if they failed to conceive. In dynastic politics, the queen was much more important than the mistress despite the mistress's more public honors because she had longevity (and the public) on her side. The alliance formed between France and her mother country was more important to the people who arranged the marriages than who was paid more attention and had more influence at court.
It's also important to remember that "prime minister" in monarchical France didn't mean exactly what it does in, for instance, modern Britain. In the context of eighteenth-century French politics, the prime minister would be the king's first minister, not the head of the government: an influential and powerful figure, but not "running the country" powerful.
France was pretty sexist, even at the time, no? The Salic Law and whatnot. So why was a woman that was not even a royal consort, and (sometimes) not even of noble birth, allowed to act almost like a PM?
This is a question I've answered a number of times, usually in reference to female monarchs in non-Salic-Law states, which is a slightly different situation, but I think this quote from one of them is useful:
The thing about sexism (and every -ism) is that it's like an iceberg: 90% of it is underwater. That is, most sexism doesn't come in the form of a man standing in front of a woman, saying, "You can't do this! This is for men only!" or blatantly sexually harassing her or believing that all men are inherently better than all women à la TRP - it's more subtle and ingrained. A lot of it comes from the assumptions people, even women, grow up with about what's seen as normal behavior.
A woman being highly visible isn't inherently a problem. A woman close to the center of power isn't a problem, necessarily - if she's promoting your candidates for officers, that's great, and if she's arranging a musical concert, that's great too. A woman close to the center of power working against you is a problem, but what are you going to do about it? The king - your absolute monarch - likes her more than you, and he's not going to cut her back just because you say women shouldn't be in charge of things. So you drop rumors about her to the press and rile up the bishops to make pointed comments about adultery at mass the next week instead.
Salic Law was certainly misogynistic, but the issue it was intended to address was women holding a masculine type of power. Unofficial influence and patronage was a completely acceptable type of power for women to wield, however.