Its murky waters, when investigating the beavers reintroduction' 500 years is fine; they are still native. The rules legislation etc are different for re introductions of native species to introduction of non native species. The levels of permissions are different this is why the despoilers are so keen to point this out. What is the time limit, I don't know, I am not sure there is any official limit but 50 years would be my maximum, after which they are non native anymore.
But should it be 50 years the other way round, once here for 50 years it is considered native, I don't know, it must depend on numbers but trying to stop invasions or getting rid of them is probably fruitless anymore. The world is too global now and that includes the spread of diseases and flora and fauna.
In light of that I don't think re introductions by man is necessary or wise whatever the reason, even deliberately spreading what we already have; enough new species are here already, why deliberately add to it; whats the point. Nature is better at sorting itself out if it is given time to do so than we can control it.
I cannot think of one "native or invaded" creature or non human disease we have successfully "deliberately" eradicated!
However, keeping their numbers down when and where necessary, finding ways of preventing the spread of diseases when they get here is the best we can hope for; I don't think we should give it up completely but those that want to make it worse....