I think the question is being approached here as an absolute. In reality a goose can HELP reduce predator susceptibility. Nothing can be guaranteed to keep away all predators. Even a Great Pyrenees can be distracted by a large pack of coyotes. If you have 2 Great Pyrenees, and they are good flock dogs, you have almost perfect predator protection. Even then a rat may sneak in the coop or a hawk may grab a bird that ranges too far from the others.
It's true that a goose will ultimately lose a battle to a determined dog, fox, etc. But geese can be more successful at preventing or stopping raptor attacks. The best way to get a goose bonded with your chickens is to have 1 goose or gander, not 2, and to raise it with a group of chicks. It's not an old wives tale that geese can help with predator protection; you just have to adjust your expectation to their reality.
Most 4 legged predators (raccoons, coyotes, possums, weasels) are primarily nocturnal. The common exception is a fox or a stray dog.
So, if you can't get a livestock guardian dog AND you don't have a problem with stray dogs (or better yet have them fenced out), you can prevent many predator losses by having a guardian goose (for daytime raptors) and a secure coop with no small entry ways.
That leaves you with foxes: if a fox starts picking on your flock and the goose isn't scaring it off, you might have to confine them for a while until you bait and catch the fox. Otherwise they may keep coming back for a daily meal until they have wiped out the flock.
Daytime raccoons and coyotes are more common in certain suburban areas where they have adjusted to humans, and where they aren't getting a lot of food in the woods. I also see more daytime raccoons during late summer when the juveniles are out exploring new territory and starting to do their own hunting. We control these by leaving live traps baited on the perimeter of where the flock forages (yes we catch our own chickens sometimes but of course release them without injury). Most of the catches are nightime racoons or possums.
Guardian dogs are the best protection, but a small yard isn't a great home for most guardian dogs. They also do a lot of barking as part of their protective instinct.