I don't understand how getting a ref to check a pitch side monitor would make any difference. It's the same group of refs who are sitting in the van making decisions, not like the ref on the pitch is going to be magically better.
VAR has just exposed how bad the standard of refereeing is; some of them can't even get it right with the benefit of replays.
1) It would be a clearer system that would get more buy-in from fans. Instead of spending a few minutes watching the ref and players mill around before VAR makes a decision based off something the ref told them that we never even hear, we would see the referee take control by watching the incident and making a call based on what he himself sees. It would be clear to everyone both in the stadium and at home who is responsible for the decision made, good or bad.
2) The main issue with VAR is the subjective decisions. In order to address this with VAR you have to accept that, while some calls are definitely right and others are definitely wrong, others are very much a matter of opinion and interpretation. The problem is that this potentially leads to one subjective opinion overruling another subjective opinion, where both are potentially valid. At which point the fans of the team that loses out could complain that the original opinion was actually no less valid than the correction. Whereas if the referee views the incident and corrects his own initial interpretation, the outcome gains more validity. The second interpretation has been deemed to be more correct even by the person who made the initial interpretation.
Which really gets to the heart of what VAR's function on tight subjective calls should be. Rather than try to come up with a definitive "right" answer (which doesn't exist), it should instead give the ref the opportunity to make the best possible subjective calls
in line with how he is refereeing the game.