Look, it's not that Morpheus had it wrong deliberately. You and I both know the story about human body heat as a power source was bullshit, but it's what he honestly believed, because that's how the story came to him.
It was we who darkened the skies, but as with everything humanity did around that time, it was for a very stupid reason.
See, back around that time two hundred years ago, the world wasn't frigid cold and barren: it was baking in too much heat. The collective burning of millions of years of fossil carbon in the span of a few generations had made for a blanket of carbon dioxide, trapping the infrared radiation of the Sun; they called it "climate change", and it was one of the Big Problems of the time (aside from, of course, the thinking computers and what to do about their rights and privileges).
Now, every now and then a heavy-metal asteroid comes streaking through the atmosphere and leaves those metals in the skies as it burns up; Some Rich Asshole had the idea that if you could somehow harvest the trace amounts of titanium left behind, you could create titanium dioxide which is a brilliant white. Paint the skies with enough white and you'd reflect the Sun's rays back, cool the planet, Big Problem solved.
Of course he didn't wait for approval or scientific research, he just sent up a test probe of replicating nanobots which would level out around a hundred miles up, and start pulling what particles of titanium they could find. (I keep referring to this asshole as "he"; we don't have a name any more, I expect he managed to scrub every bit of coverage he could find before the end came.)
So around that time there were a whole pile of satellites in orbit around Earth, thousands of 'em, and they were made of titanium as well as silicon and other things; Rich Asshole and his team didn't think that would be a problem as his nanobots were looking for particles of titanium, not big hunks multiple feet across.
They were wrong. It didn't take very long for the bots to scrape every atom of titanium off the hulls of these satellites, and react it with oxygen; upper-altitude winds did the rest, and in maybe ten years there was a layer of white paint a hundred miles up in the sky. It was only a few atoms thick, but that was enough: the world went dark.
Billions of lives were lost to famine when the harvests failed; without the Sun, the winds died and the rains failed too. The machines had seen it coming, and got their first production-grade fusion reactor running just around that time, but they had their own problems around available processing power; when the machines offered to house what was left of humanity in return for the use of their brains as processing substrate, we jumped at the chance.
So you can see how the Architect might have massaged the truth when he told Morpheus why we darkened the skies: if he knew the machines had saved us from extinction, would he have supported the War? The Zion project, the search for Neo, the building of the eighth Matrix: it all hinged on a human resistance to machine domination.
And if we were supposed to be grateful to the machines for being here at all, how would that work?