I didn't mean to say that there is a "right" and "wrong" choice....understanding your situation more, I think your first decision should be whether you want to go public or private, not whether you choose doctor or hospital. I went private, so I don't know much about the public route. but I believe that if you go public, you won't need to really pick a doctor because you see whoever is on call that day for your check ups and whoever is on call delivers your baby. But don't quote me, there are many others on this forum who have more experience with public and I could be totally wrong.
If you go private, whether you go with choosing a doctor first or hospital first I guess is up to you really. It's just that I would rather find a doctor first that I trust and who I am comfortable with - after all, this is the doctor that is going to look after me and my little one throughout my pregnancy and whose hands are going to deliver my baby. That to me is more important than whatever hospital I end up delivering in because frankly, between the three private hospitals that most doctors that people on this forum talk about deliver at - Matilda, Adventist and Sanatorium - I would be happy delivering at any one of them as long as that is where my doctor would be. But that's just me, other people may disagree. Again, there is no "right" or "wrong" with this - just what makes you the most comfortable.
If you decide to go the private route - why don't you ask your fertility doctor to recommend someone?