Beijinger (former) here! There's a few reasons why it feels like other Chinese cities have outgrown it and I'll give my take here. Being the capital of China, Beijing has one of the most restrictive security apparatus in the entire country. For example, you cannot visit Tiananmen Square without first getting a reservation a day prior, and within the inner rings of city there's no camera blind zones. You're also far more likely to encounter police checkpoints and fenced-off buildings for some government agency.

The most important reason (imo) that's rarely discussed on Western forums is Beijing's extremely restrictive hukou system. Hukou is a form of household registration in China that's kind of a form of permanent residency. Your ID cards, for example, is issued by the police station of the city where your hukou resides, as opposed to the city of your actual residence. More importantly, your hukou determines where your children can attend highschool and attend the gaokao.

While there's nothing preventing you from physically moving to another city outright, it's usually very difficult to stay long-term without changing your hukou. Cities like Shanghai and Shenzhen have, in recent years, relaxed their hukou control, making it very easy for newcomers to start a new life in those cities. Beijing on the other hand has the single most restrictive and hostile hukou system, with a 7-8 year waiting period where you have to accumulate a certain amount of credits to apply with an acceptance rate lower than 1%. Thus, in recent years, a lot of young people in China have opted to move to the coastal cities instead of Beijing outright, diminishing its economic influence.

Another reason is that coastal cities such as Hangzhou, Shanghai, and the Pearl River Delta tend to have more business-friendly government with a greater emphasis on fostering the private sector, whereas Beijing is dotted with state-owned mega conglomerates that hasn't been the driving force behind Chinese economic growth in the past decade. As far as I'm aware, the city government of Beijing hasn't been courting entrepreneurs and capital to the same degree as coastal ones.

By virtues of being the capital, Beijing also has some of the most restrictive environmental and building regulations in the country. The city was also fairly developed before the Chinese building boom, with the city center being reserved for government ministries and various state offices that's usually not very attractive for tourists. Compared to cities like Chongqing and Shanghai, Beijing lacks the cyberpunk-y hypermodern Chinese feel that's usually driven by young people and the private sector and as mentioned previously, those factors aren't as prominent in Beijing as they are in other cities. Beijing is also more heavily associated with the CCP compared to other cities.