Hello, I read your post with interest, and yes, it is a shame to be apart at the end of the pregnancy, and I hope you could make it in time to be there with your wife for the birth ..
But of course I understand...
In a public hospital , midwives help you give birth, the gyno is called just in case of a problem.
In a private clinic, you have your own gyn ( he works in the clinic) who is there for the birth.
In the vast majority of cases women get a epidural, unless they prefer to do without.
As for the health system coverage, English posters will know better than I do..
Life has a habit of biting you on the bum in ways that you least expect.............