Yes, as you said people call PRC 'China' which is already a mistake, a successful attempt to split PRC and ROC. We all fell for this.
"Great China" should be all ethnic Chinese desire. The idea of "Third Reich" started out simply as all Germany's people desire as well, even German Jews in the beginning. Then it was warped by Hitler and begin a unsatiable conquest. If Hitler's desire for his "Third Reich" stopped after he got part of Yugoslavia then there would have delay WWI many years but he pushed the line to far. If he has stopped, today on the map may even have a "Third Reich". It seems the Europe leaders and US fear him due to his large and powerful army so they sacrifice Yugoslavia to him (what politics right?) If PRC want to be "Great China", they should not repeat the mistake of the deranged Hitler. The new leaders of PRC doesn't seem to be the kind like Hitler so PRC may only use military means on ROC and not militarily occupy any other neighboring countries.
So what happen in the Arab countries, that one what to share power with another country but that country refuses? They fight. Since PRC insist ROC is part of China and within ROC, the DDP insist that they are not part of China, they will fight too. In the end, humans usually all use violence achieve their goals whether right or wrong.