Story about shapeshifting aliens trying to invade Earth, thwarted by biodiversity

I'm trying to find a short story. Here is what I remember about the plot:

• Shapeshifting aliens try to invade planet Earth. They have tried multiple times in the past, but every single time they lost contact with the attack team immediately after it landed on Earth


• The story is told from the point of view of one such alien, part of a later invasion force


• He lands on Earth, and notices the extreme biodiversity of the planet (compared to his own)


• He soon figures out that the previous teams were not dead. They just found a niche in the ecosystem in which they could be an animal (or a plant, or anything really) that had the kind of life they always wished to have



• However, this particular alien is a pilot. He loves flying. There is nothing in our ecosystem (or so he thinks) that could help him with this


• The story ends with the Earth army storming the hiding place of the alien. They find nothing, other than a large bird (a goose?) slowly flying away. It looks like there was a niche for our particular alien, after all...

I read this story a long time ago (more than 10 years), but it is probably older than that. I have no idea about the original publication date. Most likely, I read it in one of Asimov's anthologies, but I don't know if he was the author or just the publisher.

Answer

I agree with the answer above in the comments that this is most likely the story "Keep Your Shape" (also titled "Shape") by Robert Sheckley. I'm writing this answer to explain that there are (at least) two versions of this story, with two different endings. The first version was published in Galaxy magazine (Nov. 1953) and is available at the Internet Archive and at Project Gutenberg. It ends with:

Then he set out after the Hawk, which was now only a dot on the horizon. He would find out how the Hawk had broken flight as it had—skidded on air—he wanted to do that too! There were so many things he wanted to learn about flying. In a week, he thought, he should be able to duplicate all the skill that millennia had evolved into Birds. Then his new life would really begin.

He became a torpedo-shape with huge wings, and sped after the Hawk.

The second version was published in Argosy (May 1955) and seems to also be the version included in Untouched by Human Hands, Is That What People Do? and Store of the Worlds. It ends with:

The Men raced to the window and stared out. But they were unable to understand what they saw.

There was only a great white bird out there, flapping awkwardly but with increasing strength, trying to overtake a flight of birds in the distance.

The second version is I think a better match to the questioner's recollection.