Distributing free contraceptives, without requiring people to posess or use one, only increases the range of options open to people. From any kind of utilitarian standpoint, this looks like helping people to achieve what they want (if they choose to use contraceptives, they don't want children) while also achieving what you want for them to do, which is a clear win.

Of course, from an American cultural and political point of view, this wouldn't fly, because the permissibility contraceptives is a big point of blue-green politics there. Even if only a minority actually wants contraceptives to be illegal, the issue is strongly enough associated with politics that most people won't be able to think about it; they'll think about us vs. them instead. So you won't get anything done.