Some of my colleagues work closely with school nurses, who encounter some really shitty situations involving young people. In my experience, they try hard to do what's the right thing for the young person, balancing up their right to self determination within the realm of their capabilities and the need to ensure they aren't harmed. It ain't an easy job and I think it's simplistic and inaccurate to suggest that they are just handmaidens for the patriarchal medical establishment.
Readyveg, the scenario you describe in the third paragraph is ghastly, but in my experience, not uncommon. As you say, the men who abuse and exploit vulnerable women don't give a shit whether or not they're on the pill, have an implant or whatever. It's not like they'll pass a girl by if she isn't and in fact, some actually WANT them to become pregnant as this gives them a means of exerting further control over them.
So, if you deny young women in that situation access to contraception, it ain't going to make her any less vulnerable to continued abuse and exploitation. However, it will increase her risk of unplanned pregnancy (which carries more health risks for a young woman than contraceptives,) which creates a scenario "tying" her to her abuser and possibly pole axe her chances of continuing in education and strictly limits her life choices. And, I can point you to the stacks of evidence showing poorer health, education, economic, relationship, etc. outcomes for the children of very young mothers living in such chaotic, unsupported circumstances.
Frankly, it sucks that we live in a society that allows, nay encourages boys and men to abuse and exploit vulnerable young women. As I said above, just dishing out contraceptives won't change that, but at least it could help them avoid pregnancy and all the negative consequences that can bring for them.
But, I would like to see more done on trying to change the situ where men are allowed to abuse and exploit young women in this way - where young women still seem to be getting the blame and stigma for what happens to them.