If you take off your bias googles you might see that "right to continue existing" typically actually means "right to hove everyone else forced to support them".

You can exist without

- others being forbiden from expressing opinions that you find distateful

- your personal wishes being financially supported by everyone else

- being given a platform to target children with your views

- whatever facet you define yourself with being represented in every media ever made

Framing it as the "right to continue existing" in order to make "the other side" seem more extremist is exactly the kind of thing an extremist does.