One of the areas where I think there could be real potential in a national public library service is in centralisation of some functions, in particular those around stock control.
There could be a single cataloguing team, and then a small number of processing and distribution centres. With a single LMS and bibliographic database a centralised cataloguing team makes sense. What about local variations I hear you ask? In most cases, and I will probably get in trouble for saying this, I think that “local variation” immerge not from any genuine need for different records, but from under resourced libraries needing to take short cuts in the cataloguing practices. With a centralised team (properly resourced) every record could have the fullest and most accurate record, with the best authority control available. Any genuine local variations could be included on the item records.
Centralised processing and distribution would save on resource, time and money and free up staff time for reader advisory. There are exceptions though; donations and repairs. How those were dealt with would need to be looked at in detail, but I imagine that constituent libraries would want to retain those functions in house.
There is also the potential for centralised services for Human resources, and administration functions. How successful those would be depends on the management/governance structure and the contracts with the local authorities.