To expand on Compersion's points:
First, the NYT article confuses an Air Defense Zone and an Air Defense Identification Zone. An ADZ implies control of the territory or sea underneath it, and comes with shoot-down rights if the counterparty does not comply. An ADIZ is does not obligate other parties to do anything - it simply obligates the originating party (here, the Chinese Air Force) to patrol the zone, track flying objects, and identify them by communicative, electronic, or visual means.
Second, the US response is actually a non-response. Any country can send whatever it wants into an ADIZ. What China can then do is track and intercept said objects - if it so chooses to do so. A friend of mine who used to be stationed at Kadena has told me that on any given day, 80-100% of the air traffic (e.g. radar signatures) in what is now China's ADIZ is civilian traffic.
To make everyone's lives easier, China has asked civilian airliners to submit flight plans through the ADIZ in advance, so that 777s full of tourists and businesspeople don't get welcomed by J-11s sporting a full complement of A2A missiles - or, in what is a more likely maneuver, get shut out of Chinese airports. The real 'battle' here is at this civilian level, and it's one in which China has already mostly won. Other than Japan, every other nation's airlines are notifying China of their flight plans in advance. This means that China's job of sifting through air-defense radar returns just got cut by 4/5ths - which means China has already gotten what it wants.
Now, the US could certainly try to arm-twist South Korea, Singapore, Australia, and Taiwan into removing their airlines off the list, but that would be a legitimately unfriendly act towards China since China has already put its own airlines on the ADIZ lists of all those countries, and China could legitimately retaliate while successfully arguing any retaliation at the WTO.
The final piece of this puzzle is that China has built 280 airports in the past 10 years. China is the fastest growing civil aviation market on Earth. If any Asian carrier doesn't play ball on the ADIZ, it can expect to lose out on the market, since the profitability of an airline is notoriously sensitive to government regulation - even extremely 'soft'/'invisible'/'reasonable' regulations governing noise levels and flight patterns. Given that most of these Asian carriers need that growth to stay alive (due to rising fuel costs), China has essentially leveraged its internal strengths to hand the US and Japan a Hobbesian choice - force your allies to impose economic losses on their own airlines or let China get air control parity in the East China Sea (remember, Japan has held the advantage there for a long time since it was the only state with a huge ADIZ).
TLDR: China's move is something that leverages China's internal strengths well, most of the media (Chinese or Western) doesn't know shit, and the US is trying to chest-thump its way past a difficult choice while leaving said hard decisions to its East Asian allies (much as it has done in the MIddle East).