its a bit more complicated than that. Network Rail is publicly owned, and passes on its costs to the franchise holders. Without a franchise model the government will lose billions in franchise fees and taxes levied on profits.
dividends only account for 2.5% of revenue received, so private ownership and the payment of dividends doesn't impact pricing as much as you think it does.in 2016, the train operating companies collect money from the passengers via fares (£9.4 billion in 2015/16) and other forms of income (e.g. parking and catering) and spend it on running trains. They spent £2.8 billion on staffing, £0.6 billion on fuel, £1.4 billion on leasing trains, £1.3 billion on infrastructure access charges and £2.8 billion on other expenditure. They also paid out £228 million in dividends, around 1.2% of the total industry expenditure
source
However, privatisation only works where there is genuine competition, and the franchise model whereby you only have one operator goes against the theory. so in this respect, privatisation is more ideologically driven than for practicality.
Whatever happens, we really do need to invest more, and whether that is public or private is moot.