I really like the idea of MVP-voting, because it will (at least slightly) improve the communication and behavior.

I just found out one bigger problem!
As it seems there will always be only one MVP per team, which on one hand makes sense. On the other hand I know that the supports contribution is very often underestimated! Sometimes a carry may have like 14-2-8 stats and gets voted as MVP because he got alot of kills and died only a few times. Anyway the sopport was MVP because he made the kills possible by warding, healing, buffing, dewarding, etc.. Most people will vote the carry player for MVP because it's the obvious choice in terms of stats.
This will result in even less support picks (way lower chance to become MVP).
Yes, I would be one of these mad support players because many players only look at kill stats. I won't make a secret about my intention of posting this. Although my fear is good reasoned.

My suggestion to effectivly rework this system is to give 2 votes: "Most Valuable Core" and "Most Valuable Support". Both have to be casted.