Yeah, you basically nailed it. Trans healthcare for minors , if such a term would need defining would be the reduction of of permanent changes brought on by puberty. Generally, adults who started taking puberty blockers as a kid can either stop taking them later and have a (very likely mostly) normal puberty, or they can start pursuing more permanent changes, like hormone therapy.
A good chunk of the 'debate' is trying to separate normal people into 'not trans' and 'lesser.' What we want is really just comprehensive healthcare- same rules for everyone.
To your last point: trans-affirming surgeries have a vastly lower regret rate than most cosmetic surgeries, which is a very good reason to allow trans adults to choose to pursue them. The people passing these laws want a lower class. at no point does "protect trans children" ever come up in earnest.