Hardly 'British' as Scotland doesn't have them.
And other countries like France manage perfectly well without the ridiculous rules about bikes and horses on some and not others, etc.
And finally, the most British thing about our rights of way situation is that their history is so class riddled. Not only from the injustices of the Enclosure Act, the continuing appropriation of open spaces and loss of common land, through to the way parish councils were 'lent on' by powerful local landowners when the definitive maps were created in the last century.
I actually think our 'rights of way' represents the very worst of Britain, right down to the absurd 'clan like mentality' of the fractious battles between ramblers and cyclists, etc.
And another thing! The maps
Ordnance Survey Produce may be adequate, (although nowhere near as good as those produced by say, Harvey Map Services), but the very 'British' thing about OS is the way they have become a 'profit generating semi autonomous agency'. All the mapping and data they have was collected using tax payers money. Now that information is sold back to the public (even for private use) at pretty steep prices. A to photo copy a small extract for use on half a dozen handouts to illustrate a village walk for my Mum's WI, was more than the cost of buying a full set of maps.
You also need a licence to quote a grid reference or even use a spot height (not even a Bench Mark) to correlate your own survey.
Contrast this with the USA, where all this information is public information and can be used freely for non commercial use, as it 'belongs' to the tax payer. You can freely photocopy maps and charts in the US, here you will go to prison.