Just to make things clear, Lexico, "chastise" is a rather dated word for "punish", and the proverb is generally used by parents or teachers to justify their strict approach to education. In another context, like here with people or things with like, the proverb can be understood in this way "people who care (about someone/something) tend to be more protective or demanding for the own good of the person/thing they care about".
Good example. If you have seen the series "24", the president's private advisor (especially Mike in the first season, or David Palmer's brother in the third one) don't hesitate to to criticise his decisions for his own good (well, we could argue that they have interests in saving his presidency, but yet).
If on the other hand, all criticism were reational & discreetly separated from personal bonds, then one would grow up with a positive view on criticising. One fictitious example from the popular TV series would be Mr. Spok, the Vulcan, who is ready to give a cold evaluation of Captain Kirk's idea without ties to either tradition or human emotions.
Good example. If you have seen the series "24", the president's private advisor (especially Mike in the first season, or David Palmer's brother in the third one) don't hesitate to to criticise his decisions for his own good (well, we could argue that they have interests in saving his presidency, but yet).