China already has ICBMs that are capable reaching the entire USA, although in limited numbers and they'd need to be located in the more eastern areas of China, so in one sense low earth orbit hypersonic missiles aren't massive leap forwards in China's nuclear deterrent capability. But that might be a somewhat simplistic view.
Firstly, the US has leaned towards the idea of developing an anti-missile shield, and withdrew from the anti-ballistic missile treaty in 2002 to enable it to pursue this goal. It is of questionable value against the hundreds or thousands of ICBMs that Russia could launch in a full scale strike, but China is estimated to only have 20-40 ICBMs capable of reaching the US (although each with multiple warheadsFizz's answer has much more up to date and reliable data on China's ICBM capability). Allowing for malfunctions, misses and successful intercepts, it's conceivably that the US could survive a Chinese strike with heavy but far from overwhelming damage.
There is also the factor that China's existing ICBMs are at the limits of their range in hitting the US mainland, which means launching from close to the Chinese east coast, and potentially being more vulnerable to boost-phase interception from ground based systems in e.g. Korea, or sea based systems.
A delivery vehicle that can be launched to orbit the earth one or more times before choosing a target can obviously be launched from just about anywhere in China, well out of range of any boost-phase interception.
A second factor is that hypersonic missiles obviously move faster than older ICBMs, at least at some points in the trajectory (arguably any missile getting close to low earth orbit is moving at roughly orbital speed at its midpoint). Faster moving targets are harder to hit and thus less likely to be intercepted, making any USA missile defense systems less effective - although it is unknown how effective they'd be in any case.
Thirdly, no-one really knows what a technological advantage will mean in practice until it is used in anger. It wasn't obvious to anyone when the Wright brothers made the first powered flight that less than 40 years later the aircraft carrier would render the battleship obsolete. It wasn't obvious to many people how the invention of the machine gun would completely change infantry warfare. It wasn't just a faster-firing gun. It completely changed how infantry engagements were fought, and was the death-knell of mounted cavalry. When the tank was invented, most just saw it as a mobile pillbox. Very effective, but primarily in support of the infantry. But then Germany showed that it opened up a completely different kind of mobile warfare that no-one had any idea how to counter at first.
So maybe hypersonic missiles will be as relevant as 3-D cinema - gets lots of headlines for a while but makes no difference. Or maybe the added capability is itself enough to make a critical difference. Maybe it enables anti-satellite capabilities in a previously unmatched way, enabling nations to wipe out the space-based intelligence gathering, communication and GPS capabilities of an opponent. Whether it will turn out to be a difference maker like that is hard to say.
Of course there is always another factor to consider (aside from the one in the question about the military just agitating for funding). The USA is obviously moving very much in to a 'containing China' strategy, and as such it is politically necessary to talk up the threat that China poses to US interests. Russia and North Korea have previously tested hypersonic missiles (I don't know how believable NK's claims are). Other countries such as France and India are also working on such systems. Those have generated much less alarm in the US than China has, although China's test system seems to be significantly more advanced. But China is already the US's primary strategic threat, so there are political reasons to talk up the dangers.