The empires along the silk road very much did use and develop these weapons, hence the "gunpowder empires" of the Ottomans, Safavids and Mughals.

selling it off to the Europeans letting them take over the world (most of it).

Ahh, but you know how the story ends. It would not have been clear to an observer in Europe or in Asia in, say, 1550 that it would be European states that would take over the world and not the massive empires of central or eastern Eurasia.

If they did, the Europeans might never have taken as much of the Old World as they did historically.

I think you're dating the rise of European dominance in Eurasia a bit earlier than it actually occurred. By the time that European, and particularly Western European states started to gain masses of colonial territory the original gunpowder weaponry as it emerged from China had been long-since obsolete.

Moreover, it's not a mere technical dominance that sealed the deal. We can argue over when, exactly, Western European powers gained that dominance but certainly by the time Napoleon invades Egypt in 1798 and just smashes the opposing Ottoman/Mamluk forces might be a plausible end date. But was it just better weaponry that made that possible? Or was it the entire administrative, industrial and social system of Napoleonic France that allowed a conscript army with mass-produced weapons to be transported across the sea, with supplies, and to then not just defeat a local army but to also occupy and administer the territory it formerly held? The evidence of attempts at "westernization" that I'm familiar with point to the latter as being the decisive factor. Simply purchasing European arms was not sufficient to preventing European hegemony in the absence of critical socio-economic reforms.

Nor, of course, was this unique to non-European Eurasian states. Pretty much every country in Europe had to make these adjustments when revolutionary France burst onto the scene.

In other words, be wary of technological determinism. It's not a very good means for understanding how and why European powers ended up ruling such a large portion of the world.

For further reading I'd encourage you to look at John Darwin's After Tamerlane which does an excellent job of the comparative geopolitical histories of Eurasian empires.

Since gunpowder was invented in China and subsequently the cannon and the handgun, why didn't the rest of Asia (and Africa) use this technology?

They did, the Qing dynasty in particular invaded and conquered China not with a nomadic horse based army as often believed but rather with a gunpowder based infantry army not so different from the Chinese armies they were fighting. Once they conquered China, they proceeded to march into the steppes of Xinjiang with an army armed with firearm and cannons and proceeded to hand the first and last permanent defeat China inflicted on the great nomadic empires which had plagued it for 2 mellenias. The Dzungar Mongols, heirs of the tradition of Genghis Khan, were utterly crushed by the muskets and cannons of this army and the Qianlong subsequently ordered his army commanders to impose a final solution to the Dzungars in a genocide which wiped out around 80% of the Dzungar people.

The 5 great gunpowder empires were Muscovy, Qing China, the Safavids, Mughals and Ottomans. The Ottomans in particular actually adopted gunpowder from the Europeans after seeing the effectiveness of European cannons against their calvary in Hungary in the 1440s, by the 1450s they were hiring Hungarian experts to build cannons which they assaulted the walls of Constantinople with. Gunpowder revolutionized both eastern and western warfare and it was not until after the 1640s that western army began to be achieve superiority over eastern ones.