It will use tons of public sector resources and attention regardless if it's PPP or a state asset.
Just look at the recent Logan/Gateway Motorway upgrade (Logan Enhancement Project) which Transurban financed for $512M. There was least $30M of government spending directly afterwards on an adjacent road, and likely more than $100M to come.
No doubt a $7b tunnel will similarly result in $100Ms or even billions in adjacent road spending.