now we're having identical arguments against the same hateful ideology?

You literally aren't. You're facing push back because you're asking for people to ignore biology, deny their sexual orientations, and bow down to every ridiculous demand you come up with. You're facing push back because when a woman says no you people go absolutely fucking insane and threaten her with death and rape. You're facing push back because your position is fundamentally unstable and not based in reality.

Having to prove that trans people aren't predators

The fundamental problem isn't whether or not trans people are predators(though a large number are, looking at you r/actuallesbians). The problem is that ANY man whether he's trans or not can simply claim he's a woman to gain access to women's spaces. We've seen this a bunch in prisons where men will claim to be women, get into women's prison, then as soon as their prison sentence is up go right back to being men.

There was literally nothing in the gay rights movement that allowed male predators into women's spaces with such ease.

that we're not trying to trans other prople

You literally are. That's why there's an increasing number of detransitioners and will likely be many, many more in years to come. You're trying to trans little children who simply play with the "wrong" toys or like the "wrong" color. Of course when you tell little kids they can't do a thing they like unless they're the opposite sex they'll start wishing they were the opposite sex. You're a bunch of regressive assholes pushing conservative gender roles and stereotypes on people.

Of course homophobia still exists and is prevalent, but what I mean is, trans people are the main target of moral outrage now.

Yeah, the homophobia still exists, a huge amount of it coming from TRAs.