Q: Why is it that civilian politicians are given the power to nominate cadets to military academies? That seems wholly outside the purvue of legislature, or politicians in the first place.
After single-member districts "became the norm", Congress decided to "democratize and diversify the ranks of military officers" at West Point.
By 1842, single-member House districts had become the norm, with twenty-two states using single-member districts and only six using at-large multi-member districts. [Single-member district]
Congressional Nominations to U.S. Service Academies: An Overview and Resources for Outreach and Management, Page 1, Footnote 1.
Historical records indicate that the congressional nomination served to help democratize and diversify the ranks of military officers. Congressional nominations ensured that academy appointees represented all geographic areas of the United States, came from a diverse set of family backgrounds, and would not be subject to executive branch political patronage. See U.S. Congress, House Committee on Military Affairs, Military Academy, report to accompany Bill H.R. No. 367, 28th Cong., 1st sess., May 15, 1844, Rep. No. 476, pp. 14-16; U.S. Congress, House Committee on Military Affairs, West Point Academy, report to accompany Bill H.R. No. 444, 29th Cong., 1 sess., May 11, 1846, Rep. No. 660, p. 2.
In keeping with that diversity (page 5),
When a congressionally-nominated academy position is vacant, a Member of Congress may nominate 10 persons for possible appointment. As DOD service academy cadets or midshipmen who received a congressional nomination graduate, or as their appointments are otherwise terminated, a nominating Member office can make new nominations to fill any vacated positions. Typically, one appointment per DOD academy per Senator and Representative is available annually. In some years, however, a congressional office might have the opportunity to make nominations to fill multiple vacancies at an academy.
Today, the number of cadets at West Point and Colorado Springs, and midshipmen at Annapolis, is limited to 4400 for each academy. (See, 10 U.S. Code § 7442, 10 U.S. Code § 9442 and 10 U.S. Code § 8454, respectively.) About 3000 nominations may be made for each academy, but only about 1200 to 1300 appointments are made and about 1000 graduate each year.