I don't have any experience living with true stock dogs (herding dogs in general, yes, but not herding dogs who ARE herding), but I have terriers, so I have a lot of experience in obtaining solid recalls and stops when offleash. I do Rally and Agility with my dogs, but foremost I LOVE to take them offleash in safe areas - beaches, fields, trails, woods, etc. Anyone who has lived with a terrier, or multiple, knows they have their own unique challenges in regards to prey drive, independence, and how quick they react to their environment - after all, unlike herders or sporting breeds, they were bred to work independently of people, with little to no real training, which means they don't have the natural drive to check in with you, or follow along, like good working dogs "should" have. A good "working terrier" is quite the opposite. But when two months ago I called all four dogs off a fleeing fox while in pursuit, I can honestly say "I get it!".
People gawk and say "oh my gosh, you let your terriers offleash!??!". But yes......I do.......and often. Heck, they do almost everything off leash except where they are in areas that the law dictates they are ON leash, because they all were raised (with the exception of one, my oldest...who had the least original training) in such a way that it was a normal part of life. My dogs don't get out of the fenced yard and go "freeeeeedom!!!!!" like so many dogs do, they go out and they turn around and say "What are we gonna do now pardner?". That is a great feeling, and a feeling that you can't manufacture without the time it takes to develop a trusting relationship. But you can't "make" them respect you - you have to earn from them, just as they must earn it from you.
For me, and many others who don't just "live with dogs", but for whom "dogs are our lives", it goes beyond simple training. It's every interaction, every learning moment, how we manage the environment to set them up to succeed, prevent them from failing where necessary, and sometimes letting them MAKE mistakes to learn from them. It's a journey, not a six-week training class. Many people who "live with dogs", just want to dabble in things and probably don't even have the lifestyle that allows them to be all that successul - around here, many people who "try herding" go to a workshop once a month or once every two months, and say their dogs "herd". It's a far cry from somebody who lives with a flock or herd and whose dog works on a daily basis. Many of those folks are doing nothing more different than one who attends an agility workshop once per month, and "does agility".
I believe clicker training can work fabulously in a herding program - it's the philosophy that's important, not the clicker itself. Using a marker to mark appropriate behaviour (for things like the stop, the down, the recall) can really aid learning no matter area you are working in, and I know it works, because there are many dogs who have learned some control behaviours for herding with the aid of a marker. Not all, necessarily, but some. It doesn't mean the reward has to be food or toys (although I'm sure there is a time and place for it as well, like any reward), in many cases the reward would very well be to work the stock, but one might very well use food at certain stages to reward behaviours in which the stock is not immediately available as reward. I know with my terriers, I can't generally reward them with what they want - the squirrel, the fox, the bird (although they have caught and killed birds within the property with a quick jump-n-grab) - so I have to be able to find alternative means of reinforcement for them! And I don't have enough controlled access to "proof" for every type of wildlife, as I don't have a fox and bunny in my basement to bring out when I need to practice, so it's imperative that my dogs will work for external rewards. I would think that for most of these dogs you are teaching, it's true for them as well.
The fact that these dogs are trained in Rally, Agility, Obedience, etc, simply means that these dogs haven't been proofed around the distractions of livestock, and haven't worked around them before. It doesn't mean they aren't otherwise very well-trained dogs. Take my dogs and throw them in a pen with sheep, they may not be perfect right away either! Good trainers understand generalizing, and how poor dogs tend to be at it, and good trainers also understand the concept of threshold, and how to control the situation so dogs don't get too stimulated right off the bat, or if they do, how to work that dog's threshold to bring them back into thinking again.