Well the argument over whether a woman should pay may a 'red herring' in some ways, gallic, but it's not in others. Doesn't matter how 'available' something is if people can't afford it, because if they can't afford it then it's not available to them.
Being female and sexually active costs you money, in a society where you have to pay for each pill, each PAP smear, each checkup. When my pill co-pay was $80 a month, I was getting a checkup then buying a 6 month prescription. That meant I had to find $480 up front - at the time that was half my rent! Oh, sure I could have gone back to my doctor EVERY month to get a pack, thereby spreading the payments out, but oh by the way it's a $30 co-pay just to walk in the doctor's door. So now my pill costs $110 a month. That's $1,320 just so I don't get pregnant. Not counting anything like the costs of a cervical smear or other annual gynae checkups. My partner at the time wasn't paying anyone ANYTHING to have sex without fear of becoming a father.
Of course, as we became more serious it became a joint expense. A lot of my friends were horrified at the idea, btw, of it being a 'couple' expense. But frankly, why not?
There are a lot of women now who are looking at their budgets and thinking "ok, I need that $110/80/50 to buy other things more: we'll use condoms. But then the condom splits and so she tries to get the morning after pill - but if she's under 18 she can't get it at a chemists' and if the chemist isn't open on sundays/the pharmacist on duty "don't agree with that sort of thing" then she can easily end up having to travel miles and miles to another town to get the (over the counter! legal!) drug she needs. And if she doesn't get it on time, she might end up pregnant. And even with help from Planned Parenthood, abortions cost around $500. Making women pay for their pills is going to make them a LOT more expensive than the $80 I was paying - my insurance was paying more than half of that cost for me.
And without access to contraception then women's place in society (as you noted) is in serious danger. The ability to control her own fertility is a cornerstone of any women's liberation.