I think the biggest gain from controlled parking zones is the prevention of dangerous parking. This can be illustrated by the case of Boundary Brook, in the proposed Donnington CPZ Zone of East Oxford. Vehicles parked here block routes, prevent visibility, and pose a serious risk to the children at Larkrise primary school, as well as to residents and to people who use the area as a walking or cycling route.There are four year olds who cycle to school here with their parents; there are nine year olds who walk here unaccompanied.
The following photos were taken on a single trip, but are broadly representative: sometimes these specific problems aren't there; sometimes there are worse examples.
In addition to the corners, crossings, and path entries, some double yellow lines are needed to prevent parking on the stretch of road immediately outside the school gates (extension of the zig-zags would also be helpful).
Now it would be perfectly possible to put in double-yellow lines to prevent this dangerous parking without a controlled parking zone. But DYLing of all the corners and other problematic areas would involve removing perhaps 20% of the on-street parking, which is unlikely to be popular with the residents given the pressure from non-resident parking. Also, experience elsewhere suggests that without the funding from CPZ fees there will be pretty much no enforcement of any restrictions.