Maybe 10-90% of technological species do make themselves extinct by a nuclear war or an environmental catastrophe. It isn't enough. To explain the Fermi paradox needs a big killer, like an innocent looking physics experiment that sterilizes a solar system.
My point was that we could already be more than a one planet civilization. So all the eggs aren't in one basket.
To be clear: That we already could be an interplanetary culture implies that _if_ there is a big killer that explains the Fermi paradox, it is either big enough to take out a solar system -- or it is behind us.
The next stop for spreading to the galaxy would be to get outside the solar system. The good part is that the Oort cloud etc is a good training stop over. And it would also insulate humanity there from what happens closer to the sun.
(And directly a civilization is interstellar, then it really are multiple cultures (communication lag etc). Those will at least partly be "evolved"/selected for tendency to spread.)
Lots of other people since at least Fermi discussed time scales for spreading in a galaxy, it is an old discussion. If you really can't find good references comment and I'll check the usual suspects for references (centauri-dreams.org).