I think that upsetting China is deliberate.
There's already an arms race in the region, because China has been on a huge military build-up for the last ten years. They're trying to create a genuine blue-water navy, though that's still years away, as they've been going quite slowly on the aircraft carrier development. But they have built a pretty huge amphibious capacity. For a country that's not got a blue-water navy, or much history of overseas deployment, that's basically a sword aimed directly at Taiwan. They're also massively militarising the South China Sea in competition with every other nation there (and much closer) for the oil resources.
I think that the US policy is to upset China enough to get them to reign in North Korea (which everyone believes China has some power to do), without upsetting them so much as to get them really pissed off. The Chinese have got to understand that the US, South Korea and Japan have legitimate concerns about North Korea, but they've been unwilling to do much about if of late. That does seem to have been changing in the last couple of years though.
Because the other bit that you missed out is the consequences of allowing North Korea to have a fully capable ICBM force. At present Japan and South Korea both operate civilian nuclear programs, and have advanced industry, but both are happy to shelter under the US nuclear umbrella. That may change. Trump can't look any more reliable to them as an ally than he does to Europe. They've been happy to accept China's nuclear forces, because China has kept them limited and acted mostly predictably. They may not choose to accept North Korea having that power over them. China would definitely not be happy with a nuclear armed Japan - and might wish to increase its nuclear resources.
So as you can see, doing nothing may be just as bad as doing something. Hence the complicated diplomatic fun-and-games.
This is the same situation with allowing Iran to get nuclear capability. Turkey and Saudi Arabia are quite unlikely to be happy about that. The Saudis have lots of cash, and can almost certainly get access to Pakistan's nuclear program. After all, Pakistan have already sold nuclear info to Libya, North Korea, Iraq, Syria and possibly others.
As I understand it, Pakistan got their nuclear helping hand from China, to help counterbalance India. And then promptly sold it to North Korea for help with missiles. As you say, arms-races are bad.