The quote is honestly awkward to me as well, but WH Auden was a poet and you have to let go of traditional grammatical structure to read poetry in English.
As for your correction, you are perhaps missing "to": " Many a species succumbs to fate..."
To read it as two clauses of one idea:
"Fate succumbs many a species": many species go extinct.
"One alone jeopardizes itself": if someone is alone they're in danger.
The idea seems similar to a quote from Tecumseh (a famous American Indian in the United States): "A single twig breaks, but the bundle of twigs is strong."
But that's my take. Please don't try to learn English from quotes by poets.