What could possibly go wrong?
Two things that we have surely learnt from the sorry story of the past thirty years or so is that:
a) private companies suck money out of the rail industry so that can appropriate more "castles and islands".
b) private companies may sometimes appear to be customer focused but if there is no short term financial benefit they will not address any long term issue (e.g. not sorting out Sunday working in the space of THIRTY years)
The failure on b) is a
government failure, because the government made it very clear to companies bidding for franchises that they wouldn't pay the extra for sorting Sunday working out properly.
That raises a second question in the whole public/private debate. The "rail privatisation" we've had on the passenger railway for the last 30-odd years has been a halfway house between public and private. Private companies have operated the trains, but under contract to the government, required to do what the government specify, and on a budget set by government. It's not a nationalised operation, but its also a long way from truly private operation.
Network Rail has been nationalised since Railtrack was collapsed by Stephen Byers, though not formally acknowledged as such until about 10 years ago, and many of the recent disputes there and in the passenger operators have been directly linked to the constraints placed upon them by the government.
Going back to the OP, we don't have a modern comparison point to use. Other networks - Japan has been mentioned, the US deserves a look in - operate in such different contexts that any comparisons are moot. We know from looking back at the latter days of BR that it managed to be borderline profitable on some parts of the passenger network, but was still dependent on subsidy and tended to look at pricing off demand rather than increasing capacity to match. Going a bit further back, we know that the pre-nationalisation companies were quite marginal, and frequently depended on particular traffics (notably coal) to balance the books - though restrictive tariff regulation also limited their ability to respond to changes in cost base which were often government driven (e.g. working hours).
That clearly implies that any fully privatised railway operation would continue to require subsidy, and be subject to government regulation and intervention. If we look at Royal Mail, privatised in 2013, we can see a likely path for such a company. Initial optimism and share prices would be followed by profit warnings and gloom as the companies so formed struggle to sort out their long term structural issues, which would then weaken them. In Royal Mail's case, the next chapter is take over by private equity; the crystal ball is very cloudy as to what will happen in that time.