I stayed at a hospital that had "Resurrection" in its name, but that was not a service they provided.

As long as you're not doing a type of healthcare they have deranged superstitions about they can be fine. My mom got very good and effective brain cancer treatment from a hospital named for a papist saint.

I'm Catholic, I've been treated at Episcopal hospitals in my country, had family treated in hospitals from all kinds of Protestant churches. It's just not really a consideration at all, AFAIK the staff are all pretty much from random backgrounds and are unlikely to reinforce any bias that the founders may have had when the hospitals were founded a century ago. I think you probably could be skeptical if you were going there for an abortion or trans healthcare, but otherwise it's not a concern at all.

Edit: with regards to whether they should be allowed to exist: I think not, all healthcare should be nationalized of course. But it wouldn't hurt to keep some part of it as a cultural relic if we're talking about the more relatively benign institutions, the ones that have been involved in indigenous genocide should definitely become entirely secularized and scrubbed clean of any ties to remaining religious institutions that were linked to the genocides.

LGBTQ+++ people: do you trust them?

I haven't had a problem with my local one but absolutely fucking not lmao.

There was an article a while back about hospitals run by Catholic organizations that would ban doctors from offering services that violate religious rules (i.e. abortion and contraception), even if the doctors do not personally hold those beliefs. While it's probably not the case with all Catholic-coded hospitals, it's still something to be aware of. They may also pressure non-Catholic hospitals into accepting these rules as they formed mergers with them, so it's a potential issue even for hospitals that don't seem like it.

I have never heard of a muslim, hindu or buddhist hospital in "the west" though these of course exist elsewhere.

Jewish hospitals in the US were founded in large part because many other hospitals would refuse to employ Jewish doctors or treat Jewish patients in the past. I would imagine that there aren't hospitals specifically for other religions in the US because they are an even smaller minority and there hasn't been an explicit need to the same extent.

Been going to see my mom in a heavily Catholic-coded hospital. I highly doubt that many people working in the place are themselves Catholic. I also trust that they would not refuse to perform a procedure that contradicts Catholic faith.

The only reason I'm fine with existing hospitals being Judeo-Christian-coded is that hospitals as we understand them in The West have their origins in the Christian church and newer hospitals tend to lack this coding. Just today, I was wandering through the halls and stumbled across the "spiritual healing room" which contained...how do I put it? Artifacts? Icons?...of multiple non-western faiths.

I guess in the end, whether you profess a religion or not, humans have spiritual needs. Somebody going through the sort of thing that requires visiting a hospital is likely to have greater spiritual needs.

Where I live hospitals are publicly owned and not religious coded. There used to be some niche for healthcare associated with religion, I was born in the 1980's at a maternity clinic run by Catholic nuns but I haven't heard of any of these type of places surviving to the present day.

Living in a place with a Lutheran state church, all hospitals has a hospital priest attached. They seem not to be dicks about it and don't accost people who doesn't ask for it. We personally had a very good experience with a hospital priest who helped us through a traumatic event a few years ago. Hospitals are also allowing other faiths to organise religious services and send priests, imams and rabbis to visit patients.

All other things being equal, I would absolutely prefer secular institutions. Before I realized I was trans it would have been more of a preference but now I am legitimately scared.

if you were choosing to carry a pregnancy would you have doubts about going to such a place when the time came?

If you can carry a pregnancy/are pregnant I would absolutely recommend avoiding religious hospitals. I have spent far too long with people like that.

I remember hearing about Salvation Army being anti-queer in hiring. Are they generally allowed to discriminate in accordance of their religious bigotries?

Yea basically. There are some amount of legal protections, and even if there weren't, they wouldn't care. A friend told me about this exact situation, HR blatantly telling them they wouldn't hire trans people.

The main hospital I've been to I don't think is actually religious-themed, but I find myself distrusting it anyway because I live in small-town, Midwest. My psychiatrist doesn't know I'm queer because I fear his reaction. The hospital assures you that they will not discriminate, but it has "My I don't discriminate shirt is raising a lot of questions the shirt should answer" vibes. Also I have a fairly stigmatized mental illness and have had inpatient stays because of it, so I definitely don't trust them having seen that side of them, but I feel that applies to pretty much any hospital. Long story short, I don't really trust hospitals in general, not just religious ones.