Tongzhi is the emperor. He is too young to manage his own affairs. He needs someone to handle his affairs. Cixi is the biological mother of Tongzhi. In the public mind, a mother's love for her son is the most authentic and selfless, so she is the most suitable candidate.
Originally, there was ancestral training, and women were not allowed to participate in politics in the Qing Dynasty, but this was only verbal and there was no clear stipulation. Cixi listened to the political situation, which is understandable. Moreover, Cixi made it clear that when the emperor came of age, he would immediately return the power. In reality, when Tongzhi came of age, Cixi regained power, and when Guangxu came of age, Cixi also regained power. No matter how reluctant she is, whether it is true or not, formally, Cixi keeps her promise.
In short, Prince Gong has no reason to launch a coup to bring down Cixi.
See if Prince Gong has the actual ability to launch a coup.
Prince Gong is single-minded, known as Ghost Six, with a smart mind and strong ability to handle affairs. Compared with Cixi, his political skill and political ability are equal. If Prince Gong was bent on mastering the Qing regime, it would be difficult for Cixi to bring it down through a coup. On the contrary, Cixi mastered the Qing regime, and it was difficult for Prince Gong to bring down Cixi. In reality, Cixi is in power, and it is unlikely that Prince Gong will launch a coup to bring down Cixi.
After the coup, Prince Gong was appointed as the deliberation king. He boldly used the Han people as ministers and launched the Westernization Movement. There is a momentum of ZTE in China. His light overshadowed Cixi. Cixi was very uncomfortable, so she cut off Prince Gong on trumped-up charges and stayed in office until the end. Cixi did it easily, but Prince Gong was unable to fight back. Finally, Prince Gong bowed to Cixi in tears, only to be reinstated, and the widowed baron did not recover.
Cixi has sex. At this time, Prince Gong realized that his ability was limited and wanted to launch a coup. There is no possibility of success.
Finally, see if Prince Gong's coup has a worldview basis.
During the Daoguang period, Prince Gong had the opportunity to be the Crown Prince. As a result, Daoguang chose Xianfeng. In Daoguang's will, he was made a prince with one heart, which means to help Xianfeng govern the country with one heart. As a result, Xianfeng died and did not take any position as arranged. Yixin did not fall because of this, nor did she have hostility with Xianfeng.
After the coup in mind, Prince Gong got the position of deliberation king, which is equivalent to the current Prime Minister of the State Council. In this position, he is conscientious and does a good job. We can't say that Prince Gong doesn't value imperial power, but by contrast, he values doing things more than being an official. He is very similar to dourgen in this point. Dourgen and Haug competed for the throne. In order to avoid infighting, maintain the strong momentum of the Qing army and seize the best opportunity to enter the customs, he made concessions, gave up the throne to Shunzhi and became the regent himself.
When Prince Gong was the king of deliberation, his goal was to achieve rejuvenation, not to master the supreme power of the Qing Dynasty.
To sum up, Prince Gong's coup is neither justified nor sure, and his ambition is not here, so there can be no coup. The coup is very risky. Once it fails, the result will be miserable and worthless. Smart Yixin won't do that.