Complementarianism neither stands nor falls on EFS. Those who think it does have confused the essence with an accident and raise questions whether they even know what the doctrine is.
As I have observed popular debates about complementarianism over the years, I have noticed how people often confuse what the doctrine is with other associations that have little or nothing to do with the teaching. In short, folks confuse the essence with the accidents.
What do I mean by confusing essence with accidents? An essential property of any object is a property that it must have, while an accidental property of an object is one that it happens to have but that it could lack (source). You may bite into an apple that happens to have a worm in it, but you would be painfully mistaken if you were to conclude that worms are part of the essence of an apple. The apple may be worse for the wear because of the worm, but an apple is an apple with or without the worm. Likewise, you may bite into an apple dipped in caramel. In that case, you can be sure that the apple has been greatly improved. But still, you know that the improvement is an accidental property of that particular apple. The apple is an apple with or without the caramel.
A similar dynamic is in play when we think about biblical doctrines. For example, I am a pastor in a Baptist church. Many Baptist churches across the country have Sunday School every week. Mine does, in fact. But it would be a mistake to conclude from that fact that Sunday School is a part of the essence of being a Baptist. A church can be Baptist with or without Sunday School. Being a Baptist neither stands nor falls on whether Baptist churches have Sunday School. If you think otherwise, then it’s likely that you don’t know what it means to be a Baptist.
This kind of confusion seems to be at the heart of many of the debates about complementarianism that I have observed over the years. Critics of the doctrine make much of the fact that some complementarians espouse essentially literal Bible translation, conservative politics, the eternal functional subordination of the Son (EFS), etc. Other critics point to abuse or misogyny carried out in the name of complementarianism. It is a profound mistake, however, to conclude that any of those things comprise the essence of complementarian doctrine. None of those things are definitional of the doctrine in any way. Some of those things arguably may seem like improvements to complementarianism. Some of them are obviously no improvement to any doctrine. And some of them are in fact in opposition to the essence of the doctrine (e.g., abuse, misogyny). But no matter how you rate them (improvement or detriment), none of them are complementarianism. The doctrine neither stands nor falls on any of them.