So as a believing catholic, I can't be a GP, because I may have to be involved in abortion. I can't be a pharmacist or a midwife for the same reason.
I can't be a registrar because I would have to register marriages between 2 people of the same sex. My doing that makes me a part it.
I cannot to talk to colleagues outside of work who ask about my beliefs because I may then be sacked for expressing them (outside the workplace).
I can't be a baker, because I may have to produce cakes with slogans offensive to me on them. (Yes I know someone who has given up her cake making business to avoid this).
I'm wondering how my kids are going to actually earn a living.
Everytime I see a doctor or a midwife I have to explain again that no, I don't want contraception, and please don't talk about that in front of my children.
Everytime I see a midwife when pregnant I have to tell them again that, no, I don't want to kill my own child.
I cannot trust my doctor or midwife to give me and my unborn child the care we need, because I know they have killed other people's children at the same age.
Nor can I set up a hospital where these things don't happen because that would be discrimination. (Even if it didn't take public funds)
My grandparents and their fellow Churchgoers, spent lots of money building a school and coming to an agreement with the government of the day for shared funding, so their children could receive their education in a religious context.
Non-Catholic think this is unfair. Well, get together, pay up and come an arrangement with the government, like we have.
My children are still taught atheistic rubbish at their catholic school, the law says they have to be. I know many people who want to close the catholics schools and set up our own (without public funding)where we teach what we actually know to be true. (That's also wouldn't be allowed in law. )
None of us can avoid our beliefs impinging on others. That is what happens if we want to live together and share resources.
In answer to the OP.
I think these debates end up in arguments because
A) none of us really want to change what we think.
B) we have such differing world views that we don't even have a shared language to discuss it. (And our basic underlying assumptions about everything are different.)
C)we have all be hurt by 'the other side' to a greater or lesser degree, so take personally what is not meant personally.