I saw a pit bull a few months ago having a grand old game of chase with his new adopted owners. Around and around the block, in front of cars (heart stopping) and the last I saw, they were still chasing him an hour later. YEah...wonderful game to play with a young dog. Instilled a reliable response here! That pitty was young, about 6 months. IT was obviously a game he had enjoyed with humans before, he was very good at it, from the big grin and skittering feet he was having the time of his life. Hope he didn't get hit. I saw 2 very close calls.
That ^^^ it's not a game, that is a dog has little or no training that has escaped and people are chasing it. Yes the dog may think it is wonderful and having a grand time, while the people trying to catch him were in a panic. And it happens all the time as we all know and it is one of the reasons why so many people are afraid to take their dogs off leash.
I absolutely train a chase game with my pups/dogs with very clear rules. I start in a controlled fenced setting and I put in place the rules first. The game is played on my terms, it is started by me and it is ended by me and at any point I can cue the pup/dog to stop and I can take a hold of them or pick them up. Often to give them their release cue for the game to begin again. They love it.
Petie during the game, likes to stalk me from behind, I turn quickly with my arms up and out like I am going to grab him. He is often in a play bow and he runs circles around me, head up, ears touching in the back of his head, front legs spread as he runs and bum tucked..........happy dog playing a game.
There is a massive difference in what I have learned to do when training/playing the 'chase' game and what you described. Yes in the past, I have chased dogs/pups as you described. But then I trained my dogs better and turned it around like so many people do.
It is no different than the tug game or any other training, there is either a gross lack of training or half a$$ed training. When trained correctly, they are a lot of fun to play with your dog, builds a stronger relationship with them and they are a valuable training tool to boot.
And nobody would disagree that these methods/games if trained incorrectly can cause problems or shouldn't be played. But that isn't the fault of the method/game it is the fault of who didn't learn how to train it correctly in the first place and screwed it up.........no different than a lot of things.