No, it wouldn't.
> The Constitution is largely very good, but it unintentionally created a de-facto two party system. With the executive branch free from that sort of nonsense, it may be that the legislature will right itself.
Not really; the EC isn't the key feature of the electoral system that produces a two party system, and the features that are -- the fact that most elections, and particularly those for the Congress, are by plurality or majority-runoff methods -- is nowhere in the Constitution, but is the product of statute (and, largely, state statutes.)
First, because its a lot easier to achieve politically, and second, because while I've got pretty clear ideas of both the legislative and executive approaches I think would be best, I don't think its a big enough change that there is plenty of reason to be cautious in imposing a uniform national mandatory rule.