// But no one can make them wait indefinitely so that's just nonsense. Once they're adults they make their own choices. //
Firstly, yes they can be made to wait indefinitely, as you could make the decision to ban transgender care altogether; and, even if not an explicit ban, then you could take it so unseriously as to make it effectively inaccessible to all but the wealthy (see, for example, the recent BBC feature
https://www.bbc.co.uk/news/uk-england-bristol-61605588 , and even this picture is out-of-date in some regions, with waiting times for new GIC referrals in some regions measured not in years but in decades).
Secondly, one thing I had thought about adding but didn't, and perhaps should have in retrospect, is that there are three broad aspects to transgender medical care. Firstly, puberty blocking -- this is fully reversible, and proper mental health support would help guide them through it in the meantime for those who wish either to stop this or to continue on. Secondly, there is active hormone replacement -- taking testosterone or estrogen supplements, depending. Thirdly, there's surgery. Both of these latter two steps lead to irreversible changes, or at least to changes that are only partially reversible with yet more surgery later in life.
As a rule, then, the second and third steps deserve more consideration, and shouldn't be taken lightly. Ultimately, this is a decision between doctor and patient, and I don't think it's my place or anybody else's to say that the decision has or hasn't been "rushed" in any given case; but it's kind of clear that if a patient says to a doctor that they think they are trans, then if the doctor whisks them off immediately to the operating table then they're doing it wrong.