Basically, the passage you cite from Fitzgerald‘s short story is the opener of Jane Austen’s Pride and Prejudice augmented by a hunting metaphor:
It is a truth universally acknowledged, that a single man in possession of a good fortune must be in want of a wife.
You note that the meaning would be different had the author chosen a sparrow or pigeon, but a more apt comparison would be to gamebirds like quail, grouse, or dove. While these three are native to North America, the partridge is not. In 1908–9, however, some 40,000 grey partridges were imported in the hope of finally establishing an American breeding stock — solely for the sake of sport. It worked: grey partridges are now at home — and hunted — across the Northern Plains of the US and Canada.
The partridge is a popular gamebird because they are very social creatures, living together in the non-breeding season in large coveys. Flush a covey of partridges and point your gun at the sky, chances are good you’ll hit something. For dove and quail, you have to aim well.
In 1920s America, however, a patridge hunt would have likely evoked the hunting parties of the British upper classes. Thus just as Fitzgerald opens his story on a golf course, the partridge is another symbol of the leisure class.