Interesting analysis, and certainly timely given the two candidates for the U.S. Presidency ;) The trouble is, your system selects this kind of person:
This body of individuals would be the least polarizing group.
The 'least polarizing group" can be paraphrased by "the group of people who are bland and inoffensive." There's something in the lizard brain of we humans that translates "bland and inoffensive" into "weak." Thanks to that lizard-brain part of us, we'd see the result of that system as "weak leadership."
Given the anger on both sides of the aisle in U.S. politics today, one could argue that the system you illustrate will make politics even more aggravating to the characters who are angry now. To translate into current angry-words, the proposed system would select for "cucks", "RINOs" "sellouts" and "corporate liberals."
If democracy is a way to mollify civil war, then anger (aggression) is very much part of the process - so this downside will crop up. The system would be more vulnerable to (shall we say) other forms of politicking. To be frank, it'd be more vulnerable to the kind of politicized violence calculated to intimidate "the least polarizing group."
It is possible to have a groups of leaders that are "least polarizing" and yet be strong - but this type of leader is almost guaranteed to be a nationalist who displays his strength against folks who are legally barred from voting - e.g., foreigners. Since they can't vote, they can't deter a "non-polarizing" strongman.