I can't speak for the whole public sector (or private - I worked in both) and neither can you, but to claim there is no interest in finding more efficient and cost effective ways of working in the public sector is plain nonsense.
For at least 20 years - and I think earlier, from the early 1990s - there has been a constant effort to reduce costs, maximise revenues and make working practices smarter and more efficient. There was bound to be given that was a period of reducing resources (especially government grants and subsidies) and rising demand. It was also the era that began with Compulsory Competitive Tendering under the Major government, passed through the Best Value Reviews of Blair and moved into systematic service inspections by the Audit Commission and other agencies, combined with best practice and cost sharing across public sector organisations - including for instance early attempts at 'invest to save' through housing, health and care services.
It hasn't made the public sector 100% efficient (nor is the private sector) and there will be major differences of organisational practice and culture across local and central government, major public services, and with some geographical variations too - but to claim no one is interested in 'addressing' inefficiencies is cobblers!