The pattern I've observed repeated around issues of transgender health care is "ordinary levels of scientific uncertainty used to argue for the most extreme possible restrictions." Puberty blockers have long been used to treat precocious puberty, the main problems of which are social difficulty from going through puberty earlier than one's peers and a shorter adult height.
The opponents of transgender healthcare have used the minor uncertainty of prescribing this medication to a group a few years older to argue not for tighter safeguards, but for a legal ban on prescribing it for that purpose. Not letting doctors evaluate the potential risks and benefits for their patients is one thing, but not even wanting a medical trial to begin to gather more data is outrageous. There is absolutely no concern from these same groups about any risk of the medication to cisgender children, which casts doubt that the concerns are raised in good faith.