Why shouldn't he care who approached Carlo just because ALL the Family Heads were now dead?

A man in Michael's position would want to know every detail possible of what's going on with his enemies, who's behind what, etc. It would have been a careless mistake to do away with Carlo without extracting that information from him.

That aside...Michael may have already known it was Barzini who approached Carlo. But the finality, the satisfaction of having this betrayal confirmed by Carlo, and then watching Carlo die himself may have been another thing Michael was after.

Apple