The answer is, like many things, it depends. This is a huge span of time under discussion, and the political situation was frequently fluctuating, with different dynasties coming to power and shuffling the power structure and political objectives. In the broadest possible strokes, consistent and sustained population growth is usually the best indicator of economic health in pre modern times, and China was the most populous empire on earth for centuries; this can be attributed in part to the extensive network of roads and canals maintained by the scholar-gentry, in their capacities as officials and as local leaders in their communities, which allowed for the cultivation and shipment of immense agricultural yields.

However, it's important to note that to a degree, the purpose of the examinations was not simply to pick the best scholars to run the government; it was to to demand and extract the loyalty of the scholar gentry elites, by granting them the prestige and status they sought only on the dynasty's terms. In this, the examination system was effective, with Hong Xiuquan's rebellion being the exception that defines the rule. During the late dynastic rebellions, the Jiangnan scholar elites rallied under the dynasty's banner, raising their own militias, and then organizing these militias into massive private armies to crush the Taiping. The most famous of this class are the famous Zeng, Zuo, and Li, who pretty much defined the late Qing period. Even failed examination candidates, like a certain Yu Zhi, performed important functions in their communities on behalf of the dynasty. With the Taiping on the rise, Yu Zhi exhorted his fellow Jiangnan natives to attend lectures on the Kangxi Emperor's sacred edict, to give to philanthropic societies, to donate to local militias, to respect written text (which achieved a kind of talismanic power through the examinations).

However, this does not mean that the bureaucrats chosen by the examinations were necessarily good agents of the Empire; some of the top scoring candidates of the palace examination of 1761 had completely mediocre careers when they were actually in office, and corruption was rampant in periods of dynastic decay. Even those unconditionally loyal to the Dragon Throne, like Yu Zhi, would excoriate local officials for their neglect and corruption, with the warning that ruin would befall those who didn't turn their backs on their wicked ways.

Ultimately, it's hard to make a general judgement about a system thousands of years old, but it's even harder to make it in this case, since the question of whether or not the 'best' candidates for government positions were chosen requires us to weigh counterfactual scenarios that, in the premodern conditions being discussed, we are completely unequipped to consider.

I'll let other people who have more of a basis in Chinese history to address most of this, I'll concentrate on a specific aspect of the exams from an educational perspective. The examinations were heavily based on the study and memorization of a central corpus of art which would then be discussed at length during the exam. This kind of test, which relies on a central base of elite knowledge, is heavily skewed towards the children of the elite, as it rewards being a member of a cultural in group over raw talent. Compare this to the SAT for example, which, if you take a close look at it, is heavily focused on basic ideas and abilities and rewards raw talent over being well read (for the most part of course, reading a lot helps with the grammar and reading sections).

This also leads to a heavy stagnation in the group of people who have been chosen by the test. As you are choosing your government workers based on their knowledge of old material, you are introducing a strong strand of ultra-conservatism through self selection. This is made even worse by the fact that studying for the test took an immense amount of time (essentially most of your adolescent and early adult life) meaning that you are relying on people with little knowledge of the outside world to be the engine of your government.

The actual competence of the officials chosen via this method varied from time period to time period of course. However, the way the exams were set up, in my view, were in large part to blame for the inability of China to move past the Confucian suspicion of merchants and economic matters in general for much of its history, really until the massive reforms of the 1980's. It was this core economic weakness and conservatism that rendered much of China's development moot once the European nations had begun to become more dynamic and also probably contributed to its general weakness vis a vis outside invaders.

In short, in my view the exams were a great idea in theory, but the way they were set up crippled the government as it self selected people who excelled at studying Confucian scholarship to be the heart of the government, leading to China's surprisingly low ability to capitalize on its developments even in better times and then its stagnation later on.