Isn't a flat response curve what a speaker designer is striving for for most accurate reproduction?
This is correct. The problem is that for the most accurate reproduction of recorded sound you want the speaker-room combination to be flat, or tuned to the house curve of your choice. Once you place a speaker in a room the anomalies of the room give you anything but a flat curve. The amount of peaks, dips and nulls created by the room can be rather astonishing.
An e.q. is a tool that can be used to electronically address some of those anomalies, usually at a cheaper cost than room treatments, with decent results. An e.q. can’t do anything with a null but it can smooth out some of the peaks and dips which can make for a more enjoyable listening experience.
As for your original question; in theory it probably could but that is not the use an e.q. was designed for.