'I believe it’s right that doctors can opt out of providing care they disagree with (primarily on practical/emotional grounds for the patient, nobody wants to be given reluctant, disapproving medical care).'
Nope.
If you have objections to certain legal medical interventions, then don't go into an area of medicine where it will come up.
No doctor should be as unprofessional as to let patient know they are judging them in any way.
As for reluctant judgemental...
I had more than one experience of going GP for pill prescription, and they certainly got their message across loud and clear. When they explained at length why another doc would have to sign prescription. They were unable to sign because of religious beliefs that meant they were totally against contraception, even signing it would be against their beliefs, and generally laying it on thick.
Rather than eg. Wait at reception just need few mins to get this printed and signed etc.
Also mum hosp doc this is maybe 20 years ago. Real problems with abortion clinic due to number of docs who would not do on religious grounds. Meaning delays etc.
So that's a no from me.
You don't go for job in offy if believe booze too awful to touch.
You don't apply for butcher job if for whatever reason won't handle raw meat.
You don't decide as doc your place is in sexual health clinic if you believe sex before marriage and homosexual sex is a sin.
And you don't go into women's reproductive medicine if you refuse to provide certain legal care.
Any turning away, come back later, delay, hint of negative judgement.
These things some esp younger can be a huge deal to book etc.
Get that. May not come back.
It happens.