I'm not "offended" by any of it. I feel bad for the child. I don't believe reality television is a healthy environment for anyone, let alone a teenager on serious dosages of hormone medication. I also think, based on how young she was when she first entered into the public spotlight, that there's a lot more than meets the eye vis-a-vis the parents' influence. If you follow her arc, it starts to look much more like they're fame-seeking stage parents rather than their curated personas that they present on TV.
It's funny that you mention mental illness, because up until it became politically inconvenient, gender identity disorder was classified as a mental illness. An XY male with a penis who believes himself to be a female is no different than a 78 pounds anorexic woman who believes she's obese. In both cases, it's the mind that's dissociated with reality, not the body. Why, then, do we recognize that the appropriate treatment for anorexia nervosa is therapy but maintain that the appropriate treatment for GID (or GD) is genital mutilation? When the mind and body don't agree, we should treat the one that's wrong.
To your main point, yes I would feel the same if the topic was bullying, mental illness, or race. I wouldn't allow my 14 year old daughter to be a reality TV or YouTube star. It's not appropriate and there are far too many pitfalls.
Disposition is different from gender, though I agree that forcing a child to behave in a way contrary to their nature is counterproductive to their overall development. However, just because a boy likes to do so-called "girly" things doesn't mean he IS a girl. My daughter might grow up liking Notre Dame football, red meat, trucks, wrestling, and power tools. She might not want to shave her legs or wear makeup. Fine. None of those things make her a male.