China is not communist. It has never claimed to be communist. (Nor had the USSR made such a claim.)

"Communist" countries are, properly termed, "socialist" states because in Marxist theory (grossly simplified) the development is Capitalist->Socialist->Communist. In a socialist state the Communist Party is intended to shepherd people along the path to communism. Once communism is achieved, there is no need for a government. As such, the very term "communist government" is an oxymoron.

So China is a "socialist state". And socialist states, in communist theory, are not about "free medical care" or whatnot, like the "social democracies" of the west (like, say, Sweden) are about. Socialism, in Marxist terminology, is a very specific thing that has nothing to do with free state services (though those may be a desirable byproduct of them). And, get this, socialist states may use capitalist tools to accomplish their ends. It's just that capitalism in a socialist state is a tool used by the state, and is also under its thumb (which is why billionaires in China fear government; government in the USA, by contrast, fears billionaires).

That being said, yes, there's huge swaths of inequality in China, and education in particular is currently being massacred by it. The government attacks inequality fitfully here and there, but there does need to be a more concerted and forceful effort for it to actually work.

(Of course, with my more anarchistic leanings, I'm pretty certain that the socialist phase is a regressive concept that will never end because the people who run socialist governments really like this feeling of being in power so won't be giving it up anytime soon.)

The best parts of Actually Existing Socialism are commodity production and a centralized state!