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.
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.