Not necessarily true. Not even probably true. The nuclear-armed USA has invaded a number of smaller and weaker countries over the past decades. China has also attacked several neighbors, though in a less balls-out manner. Armageddon did not ensue.
The scariest scenario in Asia right now is: China attacks Japan (e.g. invades the Senkakus, or even accidentally kills some Japanese soldier in one of China's frequent reckless acts of attempted intimidation, like flying dangerously close), the Japan retaliates, the China escalates, then USA intervenes, and somehow USA and China end up in direct armed conflict.
But even there, China and USA have directly fought each other in a war (Korean war) and nobody pulled out their nukes (though China didn't have them yet iirc). The USSR and USA were always invading somebody or other. The nuclear Armageddon scenario is probably a lot less likely than feared--whereas I think the possibility of a nevertheless-vey-bad war in Asia is broadly underestimated.
The only direct conflict between two nuclear powers in history was the Kargil War between India and Pakistan in 1999. This was a fairly minor conflict among two somewhat minor powers subject to intense international pressure to stop the fighting before it spread.
Nobody really knows how an armed conflict between two major nuclear powers would play out, because it's never happened. The Korean War was a completely different scenario. China didn't have nuclear weapons. The Soviets barely had nuclear weapons. The USA was essentially the only nuclear power at the time. The USSR didn't gain the ability to inflict MAD levels of destruction until sometime in the 1960s, more than a decade later.
Probably the closest we came to a direct conflict between two major nuclear powers was the Cuban Missile Crisis. At one point that conflict was prevented from going nuclear only by the stubborn insistence of a single submarine officer arguing with two of his fellows. And that conflict was barely a fight at all!
Maybe a war between major nuclear powers has little chance of going nuclear. But nobody knows. There are good reasons to think that it could be hard to avoid.
Someone attacks a Great Power, they demand lots from a small country. Another Great Power swears to protect the small country. Another Great Power will defend first Great Power. Then before you know it, the power keg explodes.
I tend to agree. You have a high number of global flashpoints where conflict already exists or is barely contained and institutional fragmentation of different kinds within great power areas. It's deeply worrying.
not necessary. A modern conflict can happen in the disputed area without anybody bombing cities of the opponent. iPhones will continue to be shipped to US from China while J-20 will be playing hide-and-seek with F-22/35 over, say, the South China Sea. A very related example - Great Britain didn't bomb Buenos Aires 32 years ago nor did Argentina strike at anything Britain's outside of the war zone.
Argentina didn't have the military capability to attack anything outside of the war zone. Great Britain simply exercised restraint in their attacks on the mainland.
that is the main point. Any country escalating a conflict outside the war zone would pay a heavy price that would make it not worth it. They wouldn't win the war that way, they'd lose it. It is not their good will, it is just their desire to win, not lose. Bombing Argentina outside of war zone would pit Great Britain against whole world - a loosing position, or not-wining at the very least. You wanna fight - take it outside and everybody is happy.