I think the problem was that most rural Local Authorities were not wishing to start spending money on bus subsidies, as hitherto these services had been provided without, or at least cross-subsidised from urban or inter urban routes with the bus companies, and invisible. They were mighty suspicious of bus company accounting, and not without good cause, as no proper (as we would understand it today) route costing system was available. There was also a view (with an echo today) that Central Government, who owned these bus companies, were trying to move their expenditure onto the Local Authorities. Whilst the 1968 Act permitted them to subsidise, it was not until the 1972 Local Government Act imposed a duty of 'co-ordination' that the local Authorities really started engaging.
The NBC started sending out these letters demanding money with menaces in 1971, and Local Authorities attempted to form a phalanx to resist. You are quite right that some authorities decided to pay up, or certainly make a contribution. Unfortunately, when you make 'either,or' type threats on a matter of principle, failure to carry out the threat would be seen as a weakness to the waverers, so the axe fell quite hard in some areas. (Never a good idea to be making business decisions solely on points of principle, and always give yourself a let out!).
Most NBC companies at that time simply did not have the capacity to make wholesale network changes to operate more economically. Bus timetables of 1970 were structurally the same as 1950, with evening and sunday services (in particular) simply cut out rather than modified. The 1971 cuts/1972 'co-ordinating' function of LAs forced the NBC to develop the Operational route costing system by 1974, the mechanics of which were transparent and understood by Local Authorities and assisted in getting all but the most parsimonious on board. At least then it was known where the losses were really being made - it was still another thing to develop profitable networks; hence MAP was developed by Midland Red and spread out to many (but not all) companies.
Whilst the Drivers' hours rule changes in 1970/1 did not help, I do not think that these had much to do with the 1971 letters and subsequent rural mileage cuts. The NBC companies were strongly unionised and I think the main problems were with long distance services and Tours/excursions, with some issues surrounding minimum rest periods between duties, particularly when changing from late to early turn weeks. Overall I think it was a reduction in flexibility rather than serious cost increases. Passenger numbers had been falling since the mid 50s, particularly off peak (refrigerators, so no need to shop daily), evening and Sundays (television and car ownership). Peak loadings were pretty buoyant (car owners often travelling by bus to/from work), and bus managers [District Traffic Superintendents in those days!] were fretting as to how to carry them all at once whilst grappling with crippling staff shortages and an ageing fleet with insufficient capital resource to replace. In the rural areas, the 1962 Agricultural Wages Act was encouraging farmers to mechanise, and village populations started to shift from land based staff to well-heeled commuters with cars. Bus crew/engineering wages costs were rapidly increasing (due to the shortage pressures) and overall revenue was going down. These two pressures passed each other in most companies by 1970, hence the 1971 letters.