Sarah Paine says that the Kremlin asked Washington if Washington was OK with the Kremlin's nuking China -- during or right after what you dismiss as "small border skirmishes".

According to Paine, Washington responded with a firm no. Paine didn't specify the targets the Kremlin wanted to nuke: I'm guessing Beijing's new (and consequently small) nuclear stockpile and nuclear production facilities rather than cities.

In the Kargil War both sides deployed multiple divisions, and the Sino-Soviet War armored units were employed. Sure, neither side came close to using nukes, but they were still at war. It just goes to show that war between two nuclear states is possible without nuclear weapons being employed.