No, the right to exist in peace is not negotiable. If it is then we're living in a world that fundamentally is built on evil principles as it makes human life without value. It gives a justification to murder, rape, and steal so long as you get away with it. Human lives must have fundamental value and with it fundamental rights such as the right to self-identification (ex. gender identity and expression), economic autonomy (free to trade in a free market), and social autonomy (ex. freedom of association).
Without those rights we get a brutal world where only the strongest decides what's good and right. We've already had that for the majority of history. I'm sure no one in the West would want to go back to those times where life was short and painful. Only someone foolish would think such conditions are desirable or sustainable (at least with our level of technology, especially in warfare).
>But if, rather than just bowing out of debates, you start to demand that everyone else just step in line with your position, and threaten and mob them if they don't... well.
False equivalence. I never stated I wanted to mob anyone. But I will say if you want to be my friend or part of my family then you do have to adhere to some moral standard I find compatible. That's not up for negotiation. Everyone would agree that ultimately if you can't agree that X person has the right to exist in Y way (ex. allowing a transwoman to medically and socially transition) then there's nothing to debate. It's a conflict between fundamental principles. There's nothing to be gained by one bowing to the other. They're antagonists of the worse sort. Better they not associate than 'debate' which usually entails more violence than non-association.
Also, criticizing someone in the public sphere isn't even equivalent to mobbing. If I send you death threats or call your boss then that's mobbing, harassment, and bullying. If I call you a dipstick on Twitter and not evade your block afterwards; it's mean words. There's a key difference I should think.