According to me whether public sector or private sector is given the duties it is not going to solve our problems. The problems are there not because of public or private sectors. It is because of our attitude. We do not look at things as issues but we look at who does what. That is the problem. If it public sector there is accountability and they are blamed even for imaginary issues. If the same things are done by private sector no body bothers about it as they have no accountability.
You have quoted the example of SBI and HDFC. I do agree with you that the HDFC is a good private sector bank. But compare the charges between the 2 and then you will understand that the HDFC charges more than double of SBI charges. As for the benefits passed on to the customers is concerned there is no differences between the two. It is quite possible that the time taken for the services may be slightly more for the public sector compared to the private sector. I am having or was having accounts in SBI, Canara Bank, Indian Overseas Bank, HDFC and ICICI banks. While in HDFC the service charges are slightly high compared to SBI I am satisfied with their service as long as have not much transactions. In SBI I had problem with one branch as it was over crowded. The branch before that and the present branch are giving very good service. Other public sector banks also I have no problem. But with ICICI Bank I had severe problems. In spite of having enough funds, they bounced my cheque and when inquired they took 3 months to find out that they have made some mistake in the accounts and did nothing for the bouncing of the cheque. On other issues also I had problem so I applied for closure of the account. For that they made me to go to them about 10 times even after meeting persons right from the branch manager. I had to threaten of going to consumer cell and finally after 6 months they closed my accounts. Both the private banks came home and pestered for opening of accounts and no Public Sector Bank ever approached me for opening of an account.
I think the service of Private or Public sector institutions depend on the individuals of the concerned offices rather than whether it is public or private.