The Indefinite Generic refers to the Definition of a species, that is, those properties that are absolutely necessary for anything to be a member. It doesn't work as the subject of any predicate that isn't definitional. But with a definitional property, it's certainly true for any member. And that's one of the reasons why your sentence is ungrammatical. Tigers would still be tigers if they weren't endangered.
Madrigals are polyphonic.
A madrigal is polyphonic. Madrigals are popular. A madrigal is popular. One other reason why the Indefinite Generic a tiger is ungrammatical with the predicate become extinct is that extinction can only happen to a species, and it means that every member of the species is dead. Any tiger is in danger of becoming extinct.