I do understand where you are coming from, Old Shep, but...
It's an incredibly hard thing to do, to admit that one is beaten by a dog, and can't see a way out. And it happens, with the best of intentions, to experienced people too. One of the reasons I support organisations such as the Dogs Trust is that they are there to turn to. Not just for the street cur, but also for good people with good hearts who find themselves unable to cope with a dog.
If they really can't cope, what happens to that dog? It may have only been a bolshy teenager but now, unfortunately, its owner having failed to handle it, it does have problems that will take some expertise to overcome - and will need an experienced new owner who can cope with that background.
If the current owner feels that they can't approach a rescue organisation, be it the Breed Soc or Dogs' Trust or whatever, what will be the outcome? Most likely the dog gets locked up, chained, goes mad, bites someone, gets PTS. Or escapes and ends up - now a worthy occupant of a kennel space - in rescue - but now a much harder problem to solve and without the owner's story to inform and help with its rehabilitation.
(Or escapes, chases sheep and gets shot, of course.)
So, as I say, I do understand where you are coming from. Too many people get a dog that looks nice or is seen as some sort of status symbol or fashion accessory, and it is galling that they then pass this animal onto a charity to sort out - and probably then go and buy the next fashion animal.

Hopefully the organisation to which they take the animal suggests what would be a reasonable donation to cover the costs of rehabilitating and rehoming

Ladygrey is hardly in this category. She has tried, and is admitting that this dog is beyond her. As far as I can see, she isn't asking for anyone else to pay for the mistakes she has made (just because one takes a dog to a rescue org does not mean one doesn't cover its costs), and is trying her hardest to act responsibly and come up with the best outcome for the dog, its future owner, her other dog, her other livestock, and herself.