Why are people who match men and women in love called "matchmakers"? How did the name "matchmaker" come from?

Wang Shifu's The West Chamber in Yuan Dynasty;

The story of a pair of lovers, Zhang Sheng and Cui Yingying, who broke through many difficulties and finally got married. There is a 20% discount on five plays. The plot is: Zhang Junrui, a scholar, met Yingying, the daughter of the late Cui, at Pujiu Temple, and fell in love at first sight, suffering from inaccessibility. At this time, Sun Feihu heard that Yingying was beautiful, so he led troops to surround Pujiu Temple and asked Yingying to be his wife. In desperation, Mrs Cui listened to Yingying's idea and promised to marry Yingying if someone could quit. Zhang Sheng was overjoyed. He wrote a book and asked Du Kui, an old white horse general, to lead the troops to rescue him. But afterwards, Cui Furen never mentioned the marriage again, only let them be brother and sister. Zhang Sheng was very disappointed. Fortunately, Yingying's maid, the matchmaker, helped Yingying burn incense on a moonlit night and heard Zhang Sheng play the piano to tell her worries.

Later, Yingying heard that Zhang Sheng was ill and asked the matchmaker to visit her study. Zhang Sheng's lovesickness was difficult to solve, so he begged the matchmaker to take a message for him. Yingying pitied Zhang Sheng and finally got up the courage to write a poem in return. Later, with the help of the matchmaker, they cheated Cui Furen, had a private tryst and made a lifelong engagement. After the old lady knew it, she took it out on the matchmaker, but it was irreparable, so she urged Zhang Sheng to go to Beijing to take the exam. Zhang Sheng and Yingying said goodbye and won the first prize six months later. Zheng Hengben, the nephew of the old lady Cui, had an engagement with Yingying, so she lied that she had been accepted as a son-in-law by Wei Shangshu, and the old lady married Yingying in a rage. Fortunately, Zhang Sheng came back in time and everything will be fine. The full name of The West Chamber is Cui Yingying's The West Chamber.

Because both Zhang Sheng and Cui Yingying rely on matchmakers, people later called those who fix men and women in love "matchmakers".