My recommendation if you feel you'll need more support post birth is a public midwife led program. There is one at the WOmens. These are great, the midwife can come to your house everyday for up to 10 days and keeps coming until about 1 month when they hand you over to child health.
Most hospitals have single rooms, I have been in a major public hospital in 2 capital cities and had my own room. The first my partner could stay, the second he couldn't BUt worth asking
I just had my 2nd public and my SIL had her 1st private for the reasons you are thinking, In the end she really just wanted to go home to her own house and start life as a mum. Her lovely room started to feel too small and not her own IYKWIM? SHe also feels a lot of financial pressure from the bills that were generated. That said I think she still would say private was right for her.

If money is an issue be aware that there are more costs with private than what you first think, like the paediatric assessment and follow up, There are also often less staff on prvate wards. Support in public is awesome in terms of physio, lactation consultants etc and you can often go back for free.

I am obviously a public girl through and through
Good luck with your decisions