Chinese eunuchs were the personal servants of the emperor and his household; being close and having the ear of the emperor gives you power. An eunuch could help you by putting in a good word; eunuchs that are particularly well trusted could abuse that trust by manipulating what the emperor hears, withholding some information or twisting others. I think this alone is enough to explain why some eunuchs have had such a big effect in history.
I don't think eunuchs were an especially destabilising institution. If you did a comprehensive survey of all the power struggles throughout history, you will find many factions at play, including the usual suspects like princes (e.g. Li Shimin), generals (An Lushan), chancellors (Cao Cao), empresses (Wu Zetian), palace guards (Emperor Taizu of Song), ethnic/religious rebellions, and invasions. Eunuchs is just one of many factors.
I think it's important to be mindful of traditional Chinese historiography and its biases. Eunuchs were vilified due to their being considered subhuman, and much like women, their accomplishments were ignored until modern times. For example, Zheng He's expeditions were ignored in official history until the 20th century.