In American universities, why are academics placed in managerial and administrative functions? I'm a new hire at an R1 university in the United States. My position focuses on the administration of research. Before coming here, I worked in private industry and government agencies - but this is my first time working within a university setting.

Here it is common for many management functions to be staffed by faculty members (either current faculty, or people who held faculty positions before moving to administrative roles). 

This has become difficult, as I've come to understand that many of peers don't have any background in the areas they are expected to be successful at. For example, academic department chairs are responsible for the financial management of their department, but most of the ones I have met with aren't currently capable of reading basic financial statements. Similarly, they may be in charge of large and costly projects, but have no established project management structure.

These lead to poor outcomes (departments that spend far more than they should, projects that run over time and over budget, etc.). I can't understand why they would be placed in these positions where it is clear they don't have the skills to succeed. 

In the management of an American university, why would academics be placed these kinds of roles rather than professionals with those skill-sets?
I'm not sure this is limited to the US as it is a fairly traditional view of what a Faculty should be. Traditionally a group of scholars shared administration as needed. When things were small, not much was needed. Some universities today, though probably mostly small ones, take this idea of Shared Governance very seriously. 

In some highly regarded colleges. department heads, and possibly even deans, are elected by the faculty from among their members and likely serve for fixed terms, possibly renewable or not. 

Such administration duties usually come with a reduced teaching load. 

Some older professors may find that the teaching and research responsibilities are becoming burdensome and want to participate in a different way. Sometimes an especially active researcher with a national or international reputation is imposed upon to take on such duties to add to the "stature" of the institution. It can be questioned, of course, whether that is a good plan. If there is sufficient clerical and professional staff this can work, but as you note it can also break down. 

In general, though, such places are willing to accept lower perceived efficiency for hoped increased collegiality. 

Some such places, but I don't have wide enough experience to say all, are also more "professor oriented" with rules and regulations more favorable to professors - teaching and advising loads, for example. This can be a good thing (or not). It can also be a good thing for students as it may also mean fewer regulations to have to deal with and a more personal relationship with the faculty. If you apply to a professor, rather than to the institution, you have the beginning, possibly of a long lived relationship.