You mention reproductive health issues, and those are a real concern. If I were a woman with a difficult problem with pregnancy, I would specifically never enter a Catholic-run hospital, because of my concern for my own life. If saving my life required termination of the pregnancy, I would not be confident that option would be offered to me. If I were a young woman, I would not seek reproductive health counseling at such a facility, due to the built-in bias of the people who own it.
If I were an LGBTQ person, I would choose a different hospital due to biases against LGBTQ people by the RCC.
Now, if I were none of those things, which is the case for me, I would have no concerns for myself, but concerns for the welfare of others would cause me to choose a secular hospital, because I am an atheist.
It might not bother you, because you are unaffected by the RCC's biases. It's important to think about others, however, and how they would be affected. Attitudes matter, whether those biases are applied or not.
The State of California and the University of California should form no relationship with a church-owned hospital. That would be unconstitutional.
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