>I think the disagreement comes from the way that activists insist that the best solution for treating [gender dysphoria, etc] is to "socially and heavy handedly force everyone to complete the illusion by treating transgender individuals as indistinguishable from their biological counterparts in every way".
Here we see that the term 'illusion' is doing a lot of heavy lifting in the background, and implies some kind of bias or preconceived notion of "real" vs "fake" or "imaginary". If this is what you get from advocates and activists, then I think you either receive this through a filter of some kind, or carry said filter within.
>That's not the only solution for treating [gender dysphoria, etc], just the current (and perhaps uniquely) American one, and it comes with a variety of problems the obvious of which stem from significant biological factors that make the illusion impossible to complete (sports, etc).
This is the treatment method that has consistently demonstrated the best outcomes for the people concerned over time. It is not uniquely American, as it was pioneered in Europe before taken to the next level in the US, among other countries/regions. Again, furtherance of the idea there is an 'illusion that must be completed'. This is not the case. Trans people are very aware just how much they do or do not fit in compared to the average population at large. No one needs to point this out to them, nor does anyone need to coddle them. By far, just being treated as how one wishes to be treated goes a long, long way.
In regards to sports, if the person in question has medically transitioned and has done so with HRT for some year(s) (I think 2 is the baseline minimum from studies?), then their overall performance in sports will be measurably less then their peers of the same gender in most instances. The few that excel are not statistically more significant than the few natural athletes who excel due to some developmental advantage (larger heart/lung capacity, etc) due to early or sustained training, genetic factors, etc. At least this is what the preliminary data is showing us so far. On average, the distribution remains about the same and on par with other athletes. This is almost always a talking point/dog whistle that is, once you peel the onion, much ado about nothing. Just like most all other contentious talking points about transgender folks, which do not vary significantly from the same points about gay rights, marriage equality, racial segregation/integration debates, or equality in voting and women's rights, etc, etc.