The sporting sense of "umpire" seems to have arrived in English in the early 18th century.
Before that it referred to someone who helped to resolve a dispute of any kind. So it is important to rid the mind of the sporting connotation when we encounter the word in Shakespeare, where it always has a general meaning of "arbitrator, mediator".
For example, in Henry VI Part 1, Mortimer refers to death as a "kind umpire of men's miseries" (II.v.29), and later in the play the King asks: "Let me be umpire in this doubtful strife" (IV.i.151).
It is not a question of our misunderstanding the meaning of the word in these cases, but of misinterpreting its force.
To think of "umpire" in its modern sense would be to treat the conflict referred to by the king as if it were a game - and it is manifestly not that.
Nor is Juliet thinking of a game when she tells the Friar, "this bloody knife Shall play the umpire" (Romeo and Juliet, IV.i.63), for her thought is of suicide.