The birth control issue is about government forcing employers to pay for their employees' birth control.
No, it's about government forcing employers to pay for their employees' health insurance. Whether the employee uses that insurance to cover birth control or treatment for the clap or hemorrhoid surgery or a flu shot is none of the employer's fsck business.
The option, of course, would be to not have employers to pay for their employees' health insurance, and go to a rational "Medicare for all" system, but no one is willing to face the corporate monster that for-profit health care has created.
All the libertarians I know adopt the attitude "Leave me the hell alone". They don't want to be "given" anything. Just the right to keep the fruits of their own labor.
In the original sense, where libertarian == anarchist, that's true. But in the American sense, after capitalists hijacked the term and founded the so-called "Libertarian Party", we have a "Libertarianism" that wants to keep the fruits of other people's labor. That is, after all, the essence of capitalism: wealth for the owners of capital, a pittance for those who do the labor. Show me an American Libertarian who wants to tear up all government-issued land deeds, resource deeds, corporate charters, copyrights and patents: all the ways the investment class accumulates wealth without labor.