Why aren't faculty applications peer-reviewed? Research articles are peer-reviewed. Funding is based on peer-review. Academic dissertations are somehow peer-reviewed (external examiner).

Everything is based on peer-review in academia. I wonder why the faculty applications are not peer-reviewed.

Disclaimer: I know peer-review process is rarely used in some systems, but my question is why it is not as common as letters of recommendation?
They are.  Applications for faculty positions are typically reviewed by a search committee, which is made up of academics.  This is a form of peer review.  

It is true that sometimes the applicant and search committee may be from the same institution when a job application is reviewed, while funding agencies and journals often do not permit that situation.  The reason is that it is not feasible to eliminate all potential conflicts of interest.  In other words, it is hard to obtain external reviewers for something like a search committee.  Also, the members of the search committee know more about their institution's needs than an external reviewer does.