That reasoning is actually wrong. Chinese dynasties had held the same attitude towards Chinese emigrants as far as we know (by official decrees) since Ming dynasty. The idea is much earlier than that. There is an ancient saying by Han Yu 韩愈 768-824AD who quoted Kong Zi (Confucius) as saying
“孔子之作《春秋》也,诸侯用夷礼则夷之,夷而进于中国则中国之。”(《五百家注昌黎文集》卷一一《原道》)
It means, treat the feodal lord (Chinese people) as foreigners if they practice foreign rites, treat the foreigners as Chinese if they act as Chinese.
Worth to note, Han Yu lives in the Chinese Tang dynasty, the zenith of Han Chinese power. He isn't some government scholars of Yuan or Qing dynasty to justify "foreign" rules. This idea was also inherited by the Qing dynasty in their policy towards emigration to South East Asia. It is an universal Chinese idea.
From this ideology, an oversea Chinese especially after generations have almost exclusively foreign mind therefor can not be Chinese and has to be treated as foreigners.
Remember, China was a super power all the way to 1845, yet it has practised the same policy for most of its history. It has nothing to do with the weakness since 1845, but very much to do with the deep concept of what is Chinese. A person must be Chinese in the mind which is impossible if the person is immersed in a foreign society. Let's be honest, no matter how hard the parents may try to keep the child Chinese, one becomes what one grows up and lives in. A drop of water can not resist the flow of the river.
One may argue against this idea according to modern western idea of nationhood, but this idea is very defining Chinese one as old as China itself, so don't expect China to give it up quickly and easily.
I myself is Chinese living abroad with Chinese citizenship. I feel the convenience of holding a foreign passport. But when it was my time to make a choice, I chose to live with the inconvenience and did not give up my Chinese citizenship. If someone makes the opposite choice for whatever reason, that just means that he/she does not value China enough, therefor China should not value that person too much either. It is not like China has to actively beg someone to remain Chinese.