Why do you think a student interested in government and politics would want to pursue a career in Legal Administration and not directly go to law school to become a lawyer? (law firm administration specifically)
As someone who spent many years working for the U.S. Congress, and was surrounded by lawyers on a daily basis, and who eschewed the law school option in favor of pursuing a doctorate, I have frequently been asked by college students what path I believe they should take with regard to graduate-level education. My answer, with rare exceptions, has been law school, and preferably a joint master's degree/juris doctor program, where three years of law school are interchanged with a two-year master's degree program. The master's degree would be chosen on the basis of the individual student's interests beyond the study of law, for example, international affairs, political science, or public administration.
Whether a student wants (and, bear in mind that "wants" change with time and experience) to practice law or not, the law degree remains the most useful option. Law school not only teaches students about the law; it also teaches them to think about problems and issues in a particular way that will remain very useful throughout their professional careers. Additionally, activities associated with the study of law, such as internships with law firms and clerkships with judges, provide invaluable experience while enhancing one's resume or curriculum vitae.
One of the most beneficial aspects of law school, beyond the knowledge one attains -- knowledge that will prove useful throughout one's life -- is the flexibility a law school degree provides beyond other academic degrees like master's and doctorates. Once an individual has attained his or her JD degree, the option of pursuing a legal career never goes away, assuming, of course, eventual passage of a bar exam. That is a nice fallback position to have irrespective of any immediate interest in working as a lawyer. I've known many professionals who left their jobs in the late 20s or 30s to attend law school precisely because it does greatly expand the students' options with respect to a professional career. In short, it is hard to go wrong with a law school degree, unless, of course, one prefers to study medicine.
Many students do not want to attend law school because their interests lie in a more narrowly focused field of study, and they very much hope to earn a degree in a field of study reflective of those interests. That is why the joint MA/JD option is attractive to many people. The student enjoys the benefit of the specialized field of study while also working towards the juris doctor degree.
An interest in politics is admirable, and lawyers are the most representative profession in elective politics, as it is among their key aides. Certainly, other professions produce politicians; I personally knew two former morticians who decided to run for office and enjoyed long careers in Congress. The interest in politics, campaigning and public policy are all more important than the field of study one pursued in his or her youth. Academic qualifications and work experience, however, will help considerably in attaining that goal.