It is about her right to privacy. As a ward of the state, she has no privacy as to medical matters. All matters become subject to state oversight and therefore there are no issues of privacy at stake.
At page 7 of the SCOFLA opinion, it cites the trial courts reasoning (the trial court's holding was upheld), and notes that requring permission from ANY other, including parents guardians, and sundry court personnel, is an impermissible violation.
Based on privacy, goes the opinion, the girl has an absolute right to an abortion, "without government interference" (page 3).
I see your argument, that a person under state control has no medical privacy rights. Maybe the court will use that rationale. But it flies in the face of a right that heretofor operates "without government interference."
Pages 37-41 provide the heart of the Florida Supreme Court decision.
What's interesting here is that the courts set themselves up as agnostic, unwilling to be the decision maker. The FLorida Supreme Court decision is that a minor can't be forced to seek court permission. So. L.G's case (the present case) pits the will of the DCF against the will of L.G. It isn't about "the will of the court," except inasmuch as the Florida Supreme Court has ruled that pregnant minors have a right to obtain an abortion without first obtaining parental or court consent.
So, the question might be phrased as "Does the DCF have more power against the pregnant minor's rights than another guardian would have?" And I add, if so, why?