All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
Offering condolences to the family of the deceased is a desirable Sunnah but if someone does not offer condolences even if he does not have an excuse, he is not sinful and it is not permissible to cut relations with him or to desert him especially if he is a relative as we clarified in Fataawa 88129, 89611 and 88155.
Therefore, the tradition that you mentioned in the question is not Islamically acceptable.
Nonetheless, if a person does not offer condolences and does not visit the family of the deceased to comfort them – especially if he is one of their relatives – then he is negligent; in this case, he should be advised and it should be pointed out to him that he is wrong, but he should not be deserted or cut relationship with.
Allaah Knows best.