there's definitiely no question that priate doctors "on average" are pushing c-section. partly it has to do with scheduling, partly it's 'what is known' but last but not least, there are some private doctors who are just good at c-sections but actually not good at vaginal deliveries.
if you have decided on the doctors (if!) then i would ask about his/her % of vaginal vs c-section deliveries. if the doctor is not experienced in v-deliveries (e.g. 10% of lower of all deliveries), then I would not discount the risk from the doctor being inexperienced in their v-delivery obstetrics.