To my knowledge, this is historically shrouded in mystery, so there is no authoritative answer. (I'd be very interested in hearing one myself.)
This page speculates (in, unfortunately, a very authoritative tone, yet with no citations...) that the negation of ある (あらない) did at one point exist, but was discarded for the antonym of ある (ない). (N.B., we do at least know that another negation of ある (あらず), did exist historically, so the first part of this claim isn't too hard to believe.)
Their reasoning goes something like...
word negation antonym
開く 開かない 閉める
大きい 大きくない 小さい
ある あらない ない
In these cases, none of the negations match the antonym ("to not open" != "to close", "not big" != "small") except for ある, and since ない is shorter, the あらない form was discarded.
I don't know if I entirely buy their argument or not. It seems logical but honestly anything could have happened historically for us to end up in our current position, so I'd personally be reluctant to actually subscribe to this theory without any evidence.