What GForce said. When you have too much g-force. G for "Ghurrghn -- get in there!"
Needle nose pliers are very wicked for that -- keep away from them. Use the broadnosed types like you said.
Twist and shove inward; brute power. For projects like mailshirts that don't really need the most artful ring cuts -- most shirts get their links cut with boltcutters or snips and end up with link ends that look mostly like ><, I horse the ends in together and overlap the points, to where they end up like /\\/ (this view is rotated ninety degrees from the other, accepting a link that ends up slightly out of round -- not so very different from how historical riveted links ended up, tending as they do to bulge where the overlap and rivet closure is.
Jewelry-type mail projects of course play by more demanding rules and can afford them. Armor-type projects are nearly as functional -- not fancy -- as a Parkerized .45 pistol. Parkerizing steel produces a rust resistant, though not altogether proof, just better than some, matte-gray finish to it. Twentieth century military. It isn't as much used any more on guns, as coating processes have advanced. It retains popularity in auto parts, how about that.
You need to exert inward pressure as you twist the ring ends back and forth so that the gap (the kerf) gradually closes. You should hear the wire ends click as they brush against each other (you can hear this is the video). Just be careful when pressing inwards that your grip on the ring is secure - if you slip you can bang your knuckles or stab yourself...