Popular election would theoretically probably just elect someone in line with the other branches anyway, if all of the elections happened at the same time.
In fact, don't Executive and Legislative branches also have simultaneous elections? I'm not sure that's working out in our favor, either. We vote for two of our branches at the same time, when the country is theoretically moving in one direction, and then one of those branches appoints the third, and this is supposed to create a system of checks and balances?
Perhaps the whole system needs a revision. Something for the modern age that prevents any single party from gaining control and power and then pushing things through.
First, I propose we formally recognize parties. Unofficially, they're already recognized, often with citizen membership. Every party that can claim a minimum amount of membership (the details of how to claim that don't need to fit into this basic napkin of a forum post) deserves representation. And not just trivial representation, but something resembling equal representation. We want to avoid a tyranny of the majority, after all, which means that even a minority party should be on the same footing as the big two. There's no reason the 5% of the country that considers themselves Green or Libertarian or Democratic Socialist should be completely ignored. The first branch would consist of a number of delegates from each part. Perhaps something similar to the electoral college we have now. A minimum number per party, increased slightly but not entirely proportionally to the size of the party. This would probably be closest to what we currently call the Legislative branch.
The second branch would be appointed by the first branch and would govern how the other branches are allowed to rule on things (like adjusting what numbers constitute a majority or how filibusters work), as well as topics that should have as much partisanship removed as possible (like foreign relations, up to and including military intervention). This would be the equivalent of our Executive branch. This branch would be much smaller, though perhaps not quite as small as the single dictator position we have now, and instead of being elected via close popular votes every four years, whoever's in the position would have to be agreed upon by every party within the Legislative branch. If they can't agree? Then it sits empty, and they don't get to pass laws. I'm gonna ballpark that it should have at least a 75% majority to elect this one.
At this point, it would be totally okay for the Executive branch to appoint the Judicial branch, which would largely function exactly as it did before. Though I think I'd add that the Supreme Court in this setup should be there for the purpose of altering the Constitution. This is what the highest court in the land should be about, setting our very basic foundational laws. The Consitution is an old document, and to reference a current topic that's potentially hitting the Supreme Court again, stuff like the ACA probably couldn't have been predicted at the time of the Constitution's writing. We need to make a ruling on what health care should look like for our country now that it's a bit more advanced then buying essential oils from the wagon that comes through once a month.