In many countries, railways have been successfully run by private organisations after de nationalisation of railways, for eg, UK, France.
In the USA, bulk of passenger traffic is handled by Amtrak, but it is a PSU.
If...
more... you expect private players to maintain the same level of operations, then NO, it would not be enough to run all services profitably. Infact, many services would be culled for want of profitability or be subsidised by govt, which would defeat the whole purpose.
Then there is always fear of crony capitalists taking advantage of the system, govt as such always behaves like a toothless tiger.
Some middle ground will need to be achieved or else, we can all keep mum on this and suffer the travesty which IR doles out in the name of service.