I am not a competitive person by nature myself, so that may be warping the lens through which I view dog sports.
Additionally, as can be seen here, not all dogs are equally trainable within a breed
https://www.youtube.com/watch?v=l6ioGb8IQlI
I can kind of understand how the specific details of one sport might give a bit of a pause when something different is encountered in another sporty...but a lot of my NON SPORT experience says otherwise.
I grew up around working farm dogs. Dogs always drove the cattle in toward the south side of the barn. New concrete was poured on the south side, so for a few weeks cattle came in on the west side. Yes, it took a little bit to correct the dog and have him bring them in on the different side, but it was no problem, and not a problem when we switched back...and not a problem a few years later when we put up a different structure for the cattle. The challenge of occasionally switching it up probably made for a smarter, more adaptable dog.
We only had a small herd of cattle, more for fun and for keeping the grove eaten down. We added some Chianina cattle. They are actually named for Chiana Valley of Tuscany, Italy but the joke was '"Chianina is Italian for 'your going to need a bigger fence'" because these cattle are some of the tallest in the world and quite 'intense'. The approach that worked with our laid back shorthorn/angus mixes did NOT work with these, and so our dog had to adapt.
Our dogs would accompany us out to the fields, and a tractor in road-gear can move a bit faster than a dog, so our dogs on their own started to run towards where they THOUGHT that we were going to get a head start when we were maneuvering from the yard to the road (slow, lots of turns, dog was much faster at that). Dogs would look back to see if they made the right call. It was clear that our smartest dog took all sorts of things into account including which tractor, equipment (Bean Bar vs Flatbed vs Sprayer, not so much recognizing the two different sprayers or a cullivator vs disk) but mainly
I think by remembering what fields we had been to and finished, and knew we generally did all the fields where we would turn south first and the fields where we would turn north 2nd, so he'd identify if it was the same tractor/equipment combo as last time (If not, reset, go to field 1) and if so, remember which field we were working at last (fields 1, 2, 3, = south, field 4 and 5, north) and if we had finished that field (were at field 3, not finished, go south! vs were at field 3, done, go north!). I am quite confident this was the system he used, because we could predict when he would predict wrong, and it was when we were breaking our pattern.
I could go on and on with stories like this, so for me I totally get where the very first time a dog who knows flyball runs agility hurdles may do it the way he always did and fowl up, but I can't imagine it would take more than a few training sessions to get him squared away, and then bring him back to the flyball for a few runs, and pretty quick he is going to be recognizing on his own 'this is the one where I get the ball and bring it back, so I need to jump the hurdles THIS way' and 'this is the one where there is no ball to fetch, so I need to jump the hurdles the OTHER way'. Pretty soon he'd be matching his style to the sport he was participating in at that specific time without missing a beat, and fluidly transitioning between them.