Hi OP, you sound like a pretty reasonable person and you've clearly said that you'll abide by the farmer/land owners request to keep your dog on a lead.
We ask people to keep dogs on a lead at all times because as you've said yourself no dog is 100% reliable. You may not know that my milking cows are in a field until you come across them, they may have a calf at foot or the bull may be with them. All of that could put you and your dog at danger as they can become aggressive when they are afraid.
You might not see your dog doing the toilet if it's not on a lead, my own dog can be sneaky if I'm not watching her. Dog mess is a huge problem in livestock farms. We grow our grass to be turned into silage to feed animals through the winter, if animals eat silage tainted by dog mess they sadly adort their calf/lamb at any stage of the pregnancy. This is pretty heartbreaking, we have lost cows who have aborted late on in pregnancy due to the complications.
Recently we've had a horrible problem with hare coursers which is both illegal and immoral. If your dog is safely on a lead you won't be mistaken for a hare courser. Most farmers are proud of the wildlife that have homes on our land.
I'm a farmer, I know many many farmers, please believe when I say the vast majority of farmers would only shoot a dog if they absolutely had too, most farmers have their own dogs and love them. MN seems to think farmers are shooting dogs willy nilly, this is honestly not the case.
I'm in Scotland too and 100% agree with the public's right to responsible access on my land. Keep your dog on a lead and you'd be very welcome here anytime. In any given day I may have 10 dog walkers going through my farm, it does get wearing if you have to ask them all to put their dogs on a lead. We've had a fair bit of abuse from members of the public who like the access but not the responsible part, that get pretty wearing too.
A genuine thanks for not giving the farmer abuse and for putting your dog on a lead. We really do appreciate it.