There need not be complex political chess boards behind this move. Putin could simply have told his former friend that his choices were (1) leave, or (2) bye-bye family.
But we're talking about brinkmanship in a decades-old personal relationship between two individuals (Prig. & Put.) whose senses of context and honour patently are warped by ordinary standards, as evidenced by their past and ongoing actions.
It seems not at all far fetched to me that, in such a strange relationship, each others personal family would be seen as sacrosanct... up to a point.
This sort of thing plays out in the corporate world too. That is, a fairly senior OG person who isn't the actual boss has a breakdown or burnout, and takes it out by sending long scathing emails-to-all-staff, complaints to directors, etc, "secure" in the assumption that they're just playing out some long standing internal issues and that things will be put right. Sometimes it doesn't quite work out as hoped, and that person finds to their surprise that their removal is the outcome.