Why “外籍球员” is written as "外援" instead of "外员"？
外援 Is short for 外來援軍 (reinforcement from outside). It refers to players who came from outside of your organization

Although in practice, all the 外援 are 外籍球员, but in theory, a team can borrow a player from another local term for a period, and that player from other term can also be called 外援 (outside reinforcement)

My point is, 外 in 外援 doesn't refer to 外籍(foreign), it refers to 外來 (from outside); and 援 is not a short form of 援助 (assistance), but a short form of 援軍 (reinforcement)

As I stated before, since all the 外援 are 外籍球员, people can safely presume 外援 = 外籍球员

outside of team sport lingo, 外援 mostly refers to "Foreign aid", short for 對外援助 or 外來援助