My pet goose (yeah, weird pet, but she was from an egg I saved and incubated) does a trick where when she sleeps she puts her head under under a wing and lifts a leg up -- she sleeps on a single leg with her head tucked like a big ball of balanced feathers with tremendous balance and grace. She tucks her leg up so high into her down fluff that you can't see it at all. She only wobbles a bit when she's deeply asleep -- never falling.
Here is a picture of a different species of goose (I have a Toulouse) doing the same behavior[0]. It's not terribly uncommon among birds.
Of course REAL one-legged birds exist, just wanted to mention that they like to confuse people once in awhile :)
[0]: https://p0.pikist.com/photos/373/410/nature-wildlife-bird-go...
This article mentions a correlation between pollution and loss-of-limbs in urban pigeons. They even mention it may due to human hair entanglement from hair salons as a primary culprit -- that seems far fetched to me, but who knows.[0]
I tend to think that it's mostly just environmental entanglement, pigeons are pretty curious animals.
[0]: https://www.birdspot.co.uk/bird-brain/why-do-so-many-urban-p...
Did you find the egg in the wild? I was under the impression commercial eggs were generally not fertilized.
On a different note, how does your goose behave toward people? My dad still tells stories of the cute easter chick he raised, which grew into a non-cute rooster that took over the back yard and terrorized everyone except my grandmother.
Anyway, in Europe or Asia one can buy 30 eggs, incubate them for a few days, and hold them up to a lightbox one by one. Probably at least one of them will be fertilized.
While building nests, birds will find and pick up man-made string, which becomes tangled in their feet, eventually cutting off circulation and killing the foot.
https://www.google.com/search?q=pigeon+string+foot
https://www.pigeonrescue.org/2018/09/11/destringing-pigeons-...