No. In practice they are the same.
Making these people out to be the same serves to turn people who might be convinced to help you into your worst enemies instead. No one likes being made out to be a bad person simply for seeking to protect what is important to them. The two are not the same, and too many people screeching that they are, and that conservatives all hate trans people is the issue in the first place. It's weird.
Which one intends to protect what they love, and what is "what they love?"
> conservatives all hate trans people
Not all conservatives, but lots of them don't want trans people to exist.
I know all about not being allowed to exist, just for being who you are, trust me, I do. I also know a lot about convincing people to LET you exist, rather than choosing between crushing your enemy, subjugating yourself to them, or running away. There is usually a better way. That seems to have been forgotten in modern days, but it certainly still exists.
It starts with following the site guidelines of arguing in good faith and assuming the best out of your opponents. In your view, "cannot exist in public view" and "cannot exist in practice" are one and the same. In their view it is not.
What you are doing is ASSERTING your reality over someone else. That's not a very nice thing to do, whether you're right or not. You WILL have pushback. If you want to convince people of the veracity of your views, you cannot hold that pushback against them. Certainly react to it, certainly defend yourself, but don't hold a grudge against people just for daring to fight you in the first place.
They might be wrong and you might be wrong. In this case, I'm inclined to agree with your view here, but that doesn't mean that people who don't are bad people. Just misguided.
If you want people to show compassion for you and your plight, start by showing compassion for them and theirs. Anything else is quite frankly a waste of time.