It definitely needs addressed and as PP have said, there's a way to do it that isn't confrontational, but just matter of fact.
In the chats you've had with him, has it been noted that the company are no longer accepting his latecoming? From what you've said it seems to have been a softer approach about why he's struggling rather than a direct conversation about his lateness being unacceptable.
If that conversation hasn't yet taken place, it needs to happen before a formal warning. You have to give him the opportunity to improve once its been made explicit that his persistent lateness is no longer being accepted.
If in the conversation he maintains its impossible to be on time then I would be as flexible as I could and offer him the chance to adapt his working hours. I wouldn't get too hung up on the fact his wife is at home and could do the school run as you honestly can't be sure of the circumstances. Instead, I'd accept that his personal circumstances meant an adjustment needs made but it is his responsibility to make sure he does his contracted hours in full.
As for the face time, reiterate at staff meeting or via email the company's policy on personal calls during work time. If it continues from that point, a formal warning.
When employees are falling short for whatever reason, expectations need to be explicit and there needs to be a degree of flexibility on both sides.
It may be that things don't change but by addressing the issue in a direct and supportive way, you're at least giving him the opportunity to alter his approach. If he chooses not to, that becomes an entirely different situation.