I think if they do the first part properly, there is no need for a referee's post match interview.Rugby do it very well, TMO is only consulted at the ref's discretion. Often he is simply checking a decision that he has already made.
The best change would be to have referees with a microphone and give a post match interview. Making the right decision is important and taking responsibility when they have made a mistake should be expected. Much more likely to consult the VAR for any questionable decision rather than the current set up of waiting to see if the VAR intervenes.
I'm not aware of the post match referee interview in egg chasing because you have already heard the full communication between ref, TMO and assistant refs and the ref's explanation to both captains. Rugby clearly has no issues or problems with its key decision making process. It has nothing to hide. Football's refusal to let us hear the decision making process makes me believe that football probably does.