Actually, the Chinese have had a serious problem designing and building quality jet engines for fighter aircraft. The two prototype "stealth" aircraft they've demonstrated (J-31 and J-20) use Russian AL-31 engines pulled from Su-27s, because the native WS-10 engines fail to pass muster consistently.
Indeed, the same is true of the J-15 (an Su-33 clone), which is currently the only carrier-capable aircraft they've flown. Fewer than 16 of these aircraft have been built. Its use of the AL-31F makes the J-15 hard to produce, but also severely limits its payload/range.
For reference, over 500 of the mainstay F/A-18E/F Super Hornets have been built, and ~1500 of the previous models (A-D) were produced.
The Russians are none too pleased about Chinese repurposing and reverse engineering of their aircraft (particularly for engines), and have stopped selling Su-27s and many aircraft parts to the Chinese over IP disputes.
The lack of Chinese airpower and carrier operations isn't a product of posturing or strategic differences. They've invested a huge amount of money retrofitting the Admiral Kuznetsov (now Liaoning), are building a second one, and are training pilots for carrier ops with foreign trainers.
The Chinese air capabilities are instead hampered primarily by a lack of engine technology and production capability. Never mind the lift-fans.