If you want to avoid or minimize tearing, massaging the perineal area is a must! I wasn't very diligent about doing the perineal massages before birth, but my OB did massage it for me for most of the time I was in the late stages of labor (in the transition & pushing stages). Even though I ended up with a vacuum suction vaginal birth, I was able to avoid an episiotomy completely and I had only 1 single stitch from natural tearing (and I'm a very petite Asian). I was very adamant about not having an episiotomy and my OB was understanding of this. If your OB makes you feel uncomfortable on this subject, I would change doctors. But like others have mentioned, you have no choice in doctors at a public hospital.
If you decide to switch to public hospital, I think you can choose to go home after a 24hr observation period if you had a pretty straightforward uneventful birth (not 100% sure on this). That way, you can maximize the time spent with baby and not worry about having a private room or rooming in. You can have the option hire a midwife to do home visits once you get home to follow up with you and baby.