What are modern examples of established economic theories being used on a large scale in real life?
Spectrum auctions are generally designed by economists using results and insights from auction theory. For example, most modern spectrum auctions are at least somewhat related to the ascending clock proxy auction of Ausubel and Cramton. These auctions are used to sell hundreds of millions of dollars of radio spectrum licenses. A similar point can be made about the auction mechanisms used by other branches of the public section (such as central bank treasury auctions).

Matching theory has been widely applied in the design of many real-world markets and allocation mechanisms. Examples include the assignment of student doctors to medical residency, the assignment of students to high schools, and the pairing of kidney donors with patients.

Compeition (antitrust) policy is based to a very large extent on results from industrial organisation. This includes policy formulation informed by many comparatively recent results on topics including bundling, collusion, and vertical restraints.