Again, culturally- and regionally-specific. My anecdotal evidence is that the people I've talked to (a narrow, biased group of people) have more nuanced views than the surveys pick up. Where I live, more people support the death penalty than are transphobic (according to polling, no less!).

Most people don't know the first thing about anything. Ask them what we should do about climate change, and they'll say "household recycling". Ask them whether they're okay with the environmental impact of a nuclear power station, and they'll say they're outraged about it. The uninformed opinion of people who haven't really thought very hard, when presented with a loaded question, is not particularly indicative of the process they use to form their views: it's more indicative of your sampling method.

I don't know much about how sanctuaries and safe spaces are run, but it's my understanding that they don't generally have "we accept anyone who meets this sentence-long description" policies. Round these parts, shelters have always been able to discriminate against trans people with reason, and they haven't been doing so, and it hasn't been an issue: public sentiment is just that. People have been medically-transitioning in English-speaking cultures for well over a century, now, and it's only been a Big Deal in the last couple of decades. Politics.