I think it's a little different in the UK. The UK has a concept of "public rights of way". These are not necessarily surfaced roads (mostly not). Public rights of way can be subdivided into "public footpaths", "bridleways", "byways open to all traffic" (BOAT), and perhaps others I don't recall. Each type has different transport mode access rights: Some are for foot traffic only, some for horses, some for bicycles, some for powered vehicles. Detailed laws on these matters may also differ between England/Wales, Scotland and Northern Ireland.
The key point of a "public right of way" is that, even where it is on private land, the public (whether local or not) have a right of access to pass and re-pass by whatever mode(s) of transport are applicable to that public right of way.
And so, if your private lane was in the UK and was a public right of way such as a BOAT, then you'd have no legal grounds to refuse access to the green laners.
Green laning is a popular thing in the UK and it does often cause conflict between land owners, so-called "local" people, and non-local green laners.
Unhappy land owners have been known to unlawfully block rights of way where they cross their land. If this occurs then users of the right of ways are entitled to remove blockages, even when on private property.
Then there are accusations of damage to green lanes by excessive vehicular usage. However, recreational green laners often point out that a lot of damage is caused by farmers using tractors on the sensitive green lanes or even by livestock, and not by their carefully driven 4x4s.
There is also the issue of inappropriate use of green lanes. For example, riding horses on a "public footpath" (which should be for human foot traffic only) or driving motor vehicles on a "bridleway" (which should be for human foot traffic, bicycles and horses only). Furthermore, conversely, because public rights of way are usually on privately owned land, the land owner can do what they want on it (as long as they do not block the right of way): Thus the land owner (or those people he or she authorises) may well legitimately, say, drive motor vehicles on a footpath or bridleway.
There is also the issue of green laning in very large numbers or driving too hard for the conditions. Plainly, even where access is legal, this can damage the surface of the green lane (which, as the name suggests, will usually be grass, soil, mud, gravel, etc.).
Lastly, I see no point paying an organisation large sums of money to be given a route that you could work out yourself, especially if your country has a similar concept to public rights of way (does it?). If your country doesn't have such a legal concept then the key function of such an organisation (which would justify their fees) should surely be to obtain permission from landowners such as yourself and to share with you some of the money they take in so as to pay for upkeep of the lane. People should obviously not trespass on private property that is not some form of public right of way.
In parallel I should mention that in some countries there are also 'public access to countryside' laws ('right to roam'). I think Scotland and also England/Wales have such laws nowadays, although I'm not up to date on it. These cover public access to large swathes of countryside, most of it privately owned. However, to the best of my knowledge, these do not give vehicular access rights; they are only about pedestrian access. Also these laws are relatively new and do not, as far as I know, supersede public rights of way where they exist.
If it sounds as if having public rights of way on your land can be a nuisance, I think that might be true. I used to live in south Wales and the valley I lived in was crisscrossed with rights of way, mostly footpaths. There was (still is) a fairly remote farm house that had several public footpaths all converge in its yard! It's not like the footpaths were ever very busy (I never saw anyone else on them despite walking a lot at the time) but, all the same, it would bug me if I lived there and knew that, in principle, strangers could legitimately turn up in my front yard at any time of the day or night.
Absolutely! They are taking money for using your land. As I note above, if they want to take money then, in order to earn it, they should be gaining the permission of landowners and sharing some of that money with you to offset any damage or needed upkeep.