In effect, almost everyone ends up going through some form of puberty by default. There are temporary effects that can be reversed at any point by the opposite hormones, but many effects are permanent and costly to adjust later. Going through male puberty at default ages makes you grow taller and sprout permanent facial hair, while you don't grow as tall with female puberty and gain breasts which are even harder to remove.

The care for trans minors seeks to delay puberty for a couple years, but this is cannot be done for too long, or the window for certain development is missed. As such, anyone who takes puberty blockers eventually has to go down either path. The consequences are that certain things cannot be reversed, so it's important to get the puberty right the first time whether that be a trans or cisgender one.

With proper therapy and informed consent to discover what the teen truly wants, it is safe and effective at preventing pain for trans kids and giving kids who decide medical transition isn't right for them an appropriate out. A member of my family explored their gender as a minor, but didn't medically transition, so I will not say people don't change their minds. However, that option for people confident in their gender at an early age does save them the horror of watching their body change without being able to stop it.

As far as why it gets made illegal, it is usually not people who are friendly to trans folks doing it. It is usually an effort to eventually ban all trans care, as saying "think of the children" gets people on board. In states like Florida that restricted care for minors, there were less advertised restrictions on adult care that occurred at the same time (preventing medicare from funding it and the like). They put their desire to not see more trans people exist first, and put empirical reality second.