A week ago, Tsai Ing-wen became Taiwan's first female president. Inevitably, people are going to judge her because of her gender. But Wu Zetian became the Emperor of China over 1300 years ago, back when women weren't very respected. Even if a woman becomes the President of the United States next year, people are still going to judge her for her gender. So with Wu Zetian, did people think she could do the job as well as a man could? Were there any people who thought a mistake had been made with appointing her to the position?
Wu Zetian did not get "appointed" into the position. When her husband Li Zhi died, he ordered that as the crown prince Li Xian was too young, Wu Zetian needed to serve as regent. By that point, Wu Zetian was essentially the Emperor - Empress? - and she was backed up by many people in power, including the Zai Xiang - chancellor - Cui Yan. The only reason these people could achieve their positions were due to Wu Zetian, and they depended on Wu Zetian to maintain their power. Since all these powerful men were so obedient to Wu, there was little for others to do, particularly since Wu was justified in her "regency".
By 684AD, within a year, Li Xian was already complaining about his lack of imperial authority under Wu Zetian's regency, and for the excuse of "disrespect" Wu Zetian deposed him and made her younger son Li Dan the Emperor. For this, Jinye of the house Xu - founding house of the Tang Dynasty - declared Wu to be a traitor to the nation, and raised an army of 100,000 men. He was then defeated and subsequently committed suicide.
To ensure that no one would "judge her", Wu set up an informer booth in front of the Imperial Palace, and ordered that anyone had the right to rat on nobles who "had treacherous ideas". She created a law that gave all informers coming to talk to her free passage, free food, and free lodgment, so that no one would be discouraged to rat. So whenever she heard malcontent from the nobility, they would be executed.
In 688AD, many kings of the house Li became afraid that Wu Zetian would crown herself as Empress and kill everyone from the house Li, so there were rebellions everywhere against her. But all of these rebellions failed.
So to answer your question, yes, lots of people were not happy. But all of them who made their malcontent public got killed, so..