I'm actually willing to go along with a few of the things that people on the right want to do to existing laws relating to 'the genders' , though outright bans on abortion are impractical and counterproductive. However since cold reasoning is hard to come by in mainstream discourse, and since the GOP itself isn't interested in cold reasoning, they don't stand a chance at actually convincing people that these laws, with their warm sounding titles, are bad.
If the GOP wanted to strategically pursue a patriarchal agenda, the most effective means of doing so would be to 1. increase female dependence on the state whilst simultaneously causing a US government bankruptcy. I myself would support such a plan though not because I'm particularly interested in patriarchy, which in my mind has a way of limiting the freedom of men. As much as it does the choices of women.
On Free birth control --- Giving out condoms might reduce incidences of pregnancy but the difference here is that the birth control [i'll call it BC for short now] is provided through a mandatory collective insurance pool. This ensures that price ceases to be an object and encourages people to sell birth control to charge otherwise ridiculous prices. BC consumers don't care what the price is because the insurance pays for it. Everyone's rates go up collectively, and we're supposed to call this "free".
And this prediction is not at all unfounded. We know that people using insurance will often choose an expensive drug over a cheaper one even if the expensive one isn't genuinely proven to be any better [and even if it is slightly, doesn't justify the extra price under normal circumstances] because the consumers decision has been corrupted with the knowledge that insurance is paying for it.
I do not buy that it is a health issue, if an insurance company thinks a particular behavior will reduce the risk of getting ill, assuming they are not legally prohibited from doing so, they will typically PAY people [or rather, discount them] for doing so.
In general we can't blame women for voting in their self interest.
If the government offers free X to group Y, group Y collectively was not necessarily up in arms trying to get X beforehand, but if another group Z opposes having to pay for X, then Z is probably going to attract the ire of group Y. The problem isn't X, Y, or Z, it's democracy.
Oh and on this:
"Yeah, welcome to living in a democratic society. You have to pay for shit you don't want. It's the price of entry."
Paying for shit you don't want is also called market failure. [though not very aptly named] This is why I oppose all forms of democracy, both consequentialist and deontological democracy.
As another aside; You can't understand gender politics until you understand how male disposaiblity features in the creation of laws and institutions, both past and present.