Let me explain from another point of view.
In "急ぎでなきゃ:ISOGI-DE-NAKYA; being not in hurry", "DE" combines "ISOGI:being in a hurry" and "NAKYA;unless".
Without "DE", it would be just a negative sentence as "急がない:ISOGA-NAI;not in a hurry". When you want to make more complex sentence with "unless", you need "DE" like as "急ぎでなければ:ISOGI-DE-NAKEREBA;unless you are not in a hurry".
Of course "でないDE-NAI" is negative for of "である
E-ARU" as you found.