I don't think the split is quite as stark as suggested in some areas, but much comes down to a lack of money, not necessarily a lack of desire. As we've just been through annual budget-setting season, it's worth recalling that a large number of the traditional "shire" counties are really struggling with the costs of social care - I can't recall the exact figures, but both Kent and Hampshire have stated that well over 50% of their budget (and I have vague recollection nearer 70%) is spent on "social care", be that for the young, the disabled or the elderly, or any combination thereof.
As such, it then comes down to a simple matter of "where can we spend what's left", and providing cross-boundary bus services then starts to fall into the "how does this benefit our taxpayers?", and is, for example, a library that is used entirely by taxpayers of county A perceived to be a better use of limited funds than a bus service that is used reasonably equally between a substantially smaller number of county B's taxpayers and even fewer of county A's? Despite the local-press style headlines, many counties do reasonably well with the limited staff and budget they have. (For all the comments, does anyone know how many staff Derbyshire CC have had working on the Hulley's fallout?).
To give a 'real life' example, Kent and Medway generally work very well together on cross-boundary services, with one authority taking the lead role and the other sending part funding, and different contracts are led by different authorities. There was a service between "South Chatham" and Maidstone ran as a tendered service on a roughly 2 hourly frequency for years, until the 2019 tendering round when the incumbent operator, not wanting to lose the contract, declared it partly commercial and combined it with an existing commercial school service, councils adding de minimis funding as a top-up for some journeys. Largely still OK, but a little awkward to use the 4pm departure from Maidstone because of scholars.
Enter Covid and on "reopening", the service remained largely withdrawn, except for the single return deminimis journey and the school journeys. No Saturday service, no sensible peak service, as the school journeys extended some 20-30 minutes out of Maidstone to a specific school. In order to get the bus back for a 3pm finish, the "lunchtime" departure was something like 12:30, so you couldn't go out for lunch (this seems surprisingly popular - you shop eat and go home, rather than rush back for lunch, or eat, shop and go home, rather than stuffing something down first). The rest of the off-peak, or the peak service was never restored, nor a Saturday service reinstated, because "not enough people are using it"... how can you use a bus that doesn't exist? When the operator eventually decided to throw in the towel, the service was put out to tender. The school journeys were barely affordable, whilst the off-peak shopping journeys were obscenely expensive, because service use had dropped off so much and the subsidy per passenger journey way beyond what any sensible council would pay.
The relevance to this thread: Medway weren't keen on subsidizing passengers leaving their area for another, whilst Kent didn't see the point in providing money for a service almost none of their residents used - and you can understand both points of view, however annoying it was. Various "people" have said it is ripe for reintroduction, but nobody quite knows how to go about it, even with BSIP. It wasn't attractive enough at 2-hourly to gain more passengers, yet wasn't viable at hourly because it didn't have enough.
