I don't think that's very convincing. Do you really think that allowing minors to access contraception isn't endorsing underage sex? It goes beyond that though, it is concretely abetting it. And without even any input from parents. The person prescribing is actually doing so without any way of knowing what the child's situation is.
But I think the more important point legally is around the idea of consent.
On the one hand, a child is too young, or too likely to be exploited, to be able to understand the consequences of sexual activity.
On the other hand, the child is mature enough to be able to undertake the decision to take drugs or use devices in order to allow sexual activity while reducing the risk of biological consequences. Which by the way always includes the real possibility of pregnancy.
They are inherently tied together because the concrete reference for both is having sex.
It's worth noting too, that one of the groups opposing this, and really I think the more important one because they are actually involved on the ground, is Planned Parenthood. We all know that they pass out puberty blockers and hormones with no questions asked. Exactly the same way they treat contraception and abortion for minors. I know personally one young person who as a teen in an abusive relationship had abortions through PP four times, and I am sure that is not a unique case. No one asked a damn thing about her situation.
I think a lot of feminist are very blind to the problem here - but clearly PLanned Parenthood is not. They don't want anyone asking too many questions about the ability of minor teens to consent.