Body leasing follows the long-standing trend in German business jargon to use English rather than German, even if the original doesn’t exist in the Anglosphere or has a different meaning. Think of Handy: where people actually speak English it's usually a mobile.
English sounds “sexier”; Arbeitnehmerüberlassung sounds like one of those cumbersome compounds invariably — and opaquely — abbreviated in German law.
The common term among native speakers of English for this employment arrangement is outsourced (to a client/customer).
English, however, is an international language not owned exclusively by its native speakers. There is nothing odd about the German business community using such terms within its own language group. Problems only arise when native speakers have no idea what you’re talking about.