I mostly see post hoc, cum hoc, and category shift fallacies supporting the claims that women shouldn't vote.
I find it interesting that many of us are quick to blame women as if they are they problem with this country (which is fairly ridiculous), but if patriarchs truly think of men as the leaders of the country/church/family unit (which I fully support), then shouldn't we realize that the greater gender despearency is the the failure of masculinity rather than femininity?
After all, the fall of Man did not actually occur when Eve ate the fruit, but when Adam followed his wife into doing so. Immediately after Adam eat of the fruit is when they both realized that they were naked. This is significant because adam, as the one responsible for his mate, theoretically could have annulled the decision that you had made per Numbers 30:3-13), but instead he failed in his role, submitted to sin, and the responsibility was on his shoulders for the fall of humanity. This view is reaffirmed in Romans 5:12 that Adam was indeed ultimately responsible for the curse on humanity.
This is exactly what is happening today. Saying that our country is failing because of feminism is actually a cop out- as men, we need to realize that our country is failing because of a lack of (biblical) masculinity. We are the ones responsible, we are the one who take the blame. It's literally our role to take the blame.
I am not contending that women should be in any form of significant political power- I am somewhat undecided on that, but leaning towards no. I am simply contending that women should be on equal voting terms as men.
Voting is a symbol of something in this country. It has come to symbolize the recognition of common validity and equal value we have as human beings, which is clearly supported in the bible (per being made in the image of God, co-heisr with Christ, etc. ) If we lived in a society where men were truly living up to the biblical standards they are supposed to be upholding, and loving and leading their wives as they're supposed to be, perhaps women wouldn't care if they voted or not- who's to say. But now, to say that women cannot vote in the society, does strongly indicate that you do not think that women are of equal inherent value as men (generally speaking).
As far as the "image of God thing" goes, in Genesis it is closely stated "in the image of God he created them male and female he created them", indicating that they are both indeed made in the image of God. However, per First Corinthians 11, it is clear that there is a hierarchy to that image. I think it is reasonable to say that man is closer to the image of god, and women are in the image of God through man, but I believe that this is a picture of hierarchy rather than greater/lesser value.
I never thought about the possibility of the "feminine leaving Adam" through his rib, but that is very interesting thought. I personally don't think it holds much weight as this idea is never stated in the bible: however, it is something to ponder.