It's a good idea in a world where that child is aware of their gender identity (which many people develop far earlier than when puberty starts) and about to start going through irreversible changes. The betrayal of their body is a big part of why trans children have such high rates of suicide.
In any case though, if you're worried about them being too young, why would you be making a stink about a medicine than exists to delay permanent changes in their body? We give it to cis children safely in the case of precocious puberty, it can be stopped and puberty will resume, and it stops a huge source of emotional pain for them.
Just because you don't need it doesn't mean that gender affirming care isn't still healthcare.
Even in this comment there's a distinct lack of recognition of the extent that hormones, and testosterone specifically, are responsible for dimorphism. People commonly think that the list of secondary sex traits is much shorter than it is and underestimate the effects of hormone therapy.
The Olympics have allowed trans athletes to compete as their gender identity since 2004 and yet trans women have not done well. One trans person has ever medaled. It was in a team sport by somebody who didn't have male puberty. There has been one trans woman competing in weight lifting, she didn't complete her lifts.
As far as I'm aware, the only sport where we have specifically studied athletic performance is distance running (https://www.researchgate.net/publication/307766116_Race_Times_for_Transgender_Athletes), where no advantage was found.
I'm also not sure I agree on what you personally define as transphobic. If you consistently other trans people and refer to them as separate from the rest of their gender, that's transphobic. For dating, there are a number of reasons why you may not want to date somebody. Genital preferences are a real thing and are absolutely a good reason not to date somebody. There are plenty of artifacts of being trans that are reasons to exclude somebody from who you date, but if the only reason you won't date somebody is that they're trans, that would still be transphobic.