This argument seems bizarre to me.
The lack of decent public transport in areas like Northumberland is surely one of the contributing factors to the divide between the South East and the rest of England. Being "rich enough to own cars" is hardly an indicator of above average wealth and arguably hasn't been so since the 1950's.
As has been pointed out many times, this area was already well served, relatively speaking, by buses. With car congestion being a major factor in reducing their usefulness.
All I can say is why don't you have a walk around Ashington one time, let alone the outlying villages, and really look. There are lots of available parking spaces in the more deprived areas.
The richer estates are all packed with two or three expensive cars, increasingly hybrids or all electrics. The reason being pretty obvious. You do need a certain level of wealth, economic security, to own and operate a car. This is why many motor vehicles in the poorer areas are dual purpose, for both personal and economic use.
Some of the richer estates now even have the luxury of getting rid of one car, to reap the even greater economic benefits of flexible working. As long as you have access to a lift or a taxi, you're good, if your nearest station lets you get right to the platform, by design.
Even in places like Ashington, there is a clear divide between those who can afford to entertain their kids for fhe six weeks summer holiday, and those who keep them at home, sometimes alone. A good way to tell the difference, is whether they own a car, let alone more than one.
The issue is not weatlh per se, it is the gap between rich and poor, both within and across regions. The ever widening gap. There being a pretty big correlation between bow rich the south east is, and the share of the national rail network they have access to. Typically as car-rail commuters.
It is only their success that means some need to resort to more space/time efficient modes, in some cases. Or being comfortably off enough to have other ethical concerns about private vehicles, such as climate or public health. Such things are not available to many in Ashington, the poorest, specifically those who could and probably should be reaping the economic benefits of a reopened rail line, rather than essentially being economically inactive (or worse).