"damage" and changes to dna is exactly how evolution works

we dont have the capabilities to irradiate the entire planet to the point of inhospitability for all life (it would require a nearby supernova)... even if we wanted, and started manufacturing nukes for that goal, we couldnt. sure we might extinguish ourselves and most animals, but some microorganism and small animal and plants are insanely resilient... enough would surely survive and repopulate after radiation wears off in a few centuries or millennia