Funding is provided for 570 hours a year only
Generally, state pre-schools are only open for these 570 hours and they exist to provide pre-school education to 3 and 4 year olds.
Private nurseries are there to provide care for children for an entire working day and have to provide the infrastructure necessary for this.
So for instance, even if you could bring a packed lunch, a private nursery still has to provide a cook to make meals for the children. Even if you only want to be there three hours a day, they still need to be open from 8 til 6. This is only cost effective if enough of the children are contributing to these costs.
The government funding rates are not enough to cover the costs in a private nurseries. If nurseries are charging back door top up charges, it's to cover costs not bolster profits.