I suspect a national popular vote system would give rise to a semi-viable third party. Whatever party ended up closest to the centre (likely the Democratic Party) would end up embracing that position and welcoming folks from the leftmost flanks of the GOP. As such, Democrats would become "the natural governing party," but Republican presidential candidates would also have a decent chance of winning due to vote splits, especially when there's fatigue towards incumbent Democratic leadership and left-wing voters don't really know where to go to keep the Republican out of office.
I don't think it'd produce "viable" third parties, but it would lead to much stronger spoilers.
IMO a Top Two Runoff or IRV (I prefer the former just because it guarantees a majority winner without the confusion of IRV) would be better than a straight plurality winner. And Top Two Runoff would be great for third parties! Everyone would hang on the endorsements of people who didn't make the runoff.
