No, even simultaneous detonation of every nuclear weapon currently in service for every nation would not render even unprotected humans incapable of life everywhere on earth, most statistics stating that nuclear weapons can destroy the earth are based on faulty observation of facts and unrealistic ideas about the rather limited danger from fallout, and just plain misrepresentation of the situation (like taking the yield of the atomic bombs used on Japan and casualties and then calculating the yield of all nuclear weapons and the "casualties" they would produce in the same way, like as if everyone lived in a city just like Hiroshima and Nagasaki) and there has already been over 2000 nuclear tests in just the last 62 years, yet you can travel basically anywhere on earth and never even know that a single bomb had been detonated without being told, there is no permanent fallout, no dangerous "dead zones" no wasteland and no mutant freaks, liberals need to stop reading comic books, they obviously have a hard time differentiating between reality and fantasy.
The total operational nuclear arsenal for the US is around 5,700 and that includes everything from the smallest tactical weapons to the biggest city-busting strategic ICBM's.
Fallout while dangerous would kill few by comparison to blast heat, shockwaves, burns, and post-exchange chaos.
A total nuclear war while a huge catastrophe, would not even come close to exterminating humanity. Heck a full nuclear exchange between Russia/US would not even kill everyone in either of those two nations, let alone the whole world.
Yes we have the capability to kill all humanity. If we gathered everyone together and had them stand out in the open in big circles while we dropped bombs on them. Then we could destroy humanity. And that's about it.
For one thing, most testing of 'the big ones' has occurred under the surface of the planet. Second, the concern is not that the fallout would kill everyone, but the ensuing 'nuclear winter'.
Archeological evidence indicates that single, massive supervolcanic eruption nearly wiped out the human race in the very distant past (Toba caldera, Indonesia, ~74,000 B.C), due to the massive amount of ash spewed into the Earth's atmosphere, circling the entire globe, severely darkening and cooling/freezing the planet for long enough to nearly cause the extinction of the human race. It's estimated that only a few hundred human females survived (perhaps even just a few DOZEN!), thus the distinct bottleneck in human geneology, associated with that date / era.
So, imagine that only many, many times more powerful, and blasts covering much more of the Earth's surface area, as opposed to one, single location. It's the nuclear winter, you seem to forget. Thick, radioactive dustclouds covering the entire planet, causing the next great mass extinction.
-That's why they say it would 'end the world'. Not because everyone is going to receive a direct hit by a warhead. Rather, life on Earth would soon become unsustainable.
But on top of that, I also think you underestimate the sheer gigatons of explosive power we're talking about, and what that kind of power could be capable of doing, were it focused on a single point, in the Earth. But regardless of that, be assured that we do indeed 'have the power to destroy all life on the Earth, many thousands of times over'. Simply because we know how much power it would take, to cause another mass extinction. Take the combined nuclear arsenals, divide their power by this, and there indeed you have the factual statement.