Back on the subject of buying a van in the UK, very many Australian visitors used to do this, and possibly still do so, creating an active market for camper vans.
This has got to be cheaper than renting privately, without the possible problems of arguing about who is responsible/pays for breakdowns or accidental damage not covered by insurance (like backing into a tree on a camp site), and getting a van which has been rented out, heavily used, and not properly maintained.
There was even a street near the High Commission in London where Australians used to buy and sell their vans. They possibly still do so, there must be some forums/blogs/whatever where these things are discussed.
Your idea of buying a car and using AirB&B may seem to be about the same cost or even cheaper than renting a van, but might not be when the cost of eating is considered. We find that when travelling in our camper van we spend less by preparing some meals in the van, instead of using restaurants. In any case, at our age, we find restaurant meals are usually far too large.
The other advantages, of course, are not being tied down to advance bookings, being free to stay anywhere without planning, and not having to keep getting on line to find accommodation and make reservations.
On some great and glorious day the plain folks of the land will reach their heart's desire at last, and the White House will be adorned by a downright moron.
H. L. Mencken 1880 - 1956
Some may not like his views, but what a prediction!