Prestige isn't particularly important when it comes to Medicine. Internships are based most significantly on other factors (for example, most of the hospitals in Melbourne depend largely on prior work experience, volunteer work, research experience and such). After that, any post-graduate study for specialisation and such would almost certainly ignore where you studied as an under-grad. And yes, all undergraduate places are CSP. However, Bond University is the (only?) private Medical uni where you can effectively pay your way in if things don't go right elsewhere.
Also, ATAR generally isn't too big of a factor in getting into Medicine so don't worry. I've got several friends sitting on 95s or so, with some even sub-90 (ERC place however). Like you said, UMAT and interview is more important. Regarding the unis, I thought there was a sticky somewhere with a list?