There's two reasons.
Historically, 無し (modern ない) wasn't the negative form of ある but an adjective in its own right (meaning 'absent', and valid for any subject). The negative form of ある was あらず. Naturally, these words meant mostly the same thing, and over time あらず was fully replaced (in Kantou Japanese at least) with ない, which later went on (in Kantou Japanese only) to be incorporated into other verbs as the negative marker -ない.
Also historically, あり (modern ある) was used for both animate and inanimate subjects. Modern いる (historical ゐる) was a verb meaning 'sit, stay' that became first an honorific and then an animate version of あり.
The necessity of using いない with animates is a result of a combination of two things: いる restricting ある to inanimates; and ない being reanalysed as the negative form of ある, invalidating its use with animate subjects. なし here is a leftover of historical uses of what became ない, and so it doesn't matter whether the subject is animate or not.
(also, いなし is not a word; the historical negative 終止形 is first -ず and later -ぬ (the old 連体形), and thus the historical negative form of ゐる would be ゐず.)
