An aging mafia boss struggles to enforce the family code of honor, as leadership switches from him to his more ruthless – greedy – unprincipled son.

Isn't this The Godfather?

Good logline. A good story about integrity.

The idea of a mafia boss dying and desperately trying to help his son to secure the legacy is very interesting and can make for a really good film.

There are a few problems with the latest draft of the logline. The biggest problem is the lack of a clear goal.

This - "Goal: To ensure his son is able to competently run the family business after he is gone." does not constitute a goal rather an ambition. The reason is that it is not something a camera can film it is an opinion the MC will have as a result of his son doing something.

How will the mafia boss know beyond doubt that his son is ready? What is the definitive moment or action that convinces him? This will be the MC's goal.

Why now? Why not 10 years ago or ten years from now? Was he given an approximate time to live? What is the inciting incident that makes him need to secure his successor?
Perhaps best to specify that after he was diagnosed with terminal cancer he needs to?
In this way being told he is dying made him take action.

"...struggles to deal with?" is one of the more commonly used descriptions of action in loglines that don't help a concept. Reason is all characters struggle to deal with something or other in their own unique way but what makes a character interesting is the unique set of circumstances he or she is in and the unique action they take.
Therefore what is it the mafia boss actually does with or to "...his treacherous relatives?"? Kills them? Gets them arrested? Has them sent to Siberia?
But If one specific family member is the antagonist specify him or her and make them the obstacle.

On that note are the relatives important enough to the plot to be mentioned in the logline? If the story is about the mafia boss grooming his son then the relatives are juts a complication B or C plot at best and don't need to be mentioned in the logline.

There appears to be little inner journey for the MC in the logline whilst not crucial it may help distinguish this from the many other mafia movies that have been made already. What is the MC flaw?

Hope this helps.