Even the major global powers failed to actually eradicate all humans (on the other side) during WW2, with actual nuclear weapons. Pretty sure cockroaches are actually easier.

It’s remarkably hard to actually eradicate all humans.

Nuclear weapons aren't actually that useful unless you're going for a nuclear winter.

You don't have to eradicate them all.

Wait for the robot workforce to be deployed, then visit a few deadly pathogen storage facilities. Society collapses and then you send in the murder bots, or you just ignore the ants.

Humanity has wiped out lots of species without deliberate organised effort.