If 2016 has lessons to provide posterity, one may be that picking the President should be a bit less democratic than it currently is. What if . . . .
1. People in the respective states vote for electors to represent them in the electoral college (remember that thing?!) once every four years:
a) Each state has to select their electors in as representative a way as possible – i.e., no “winner take all” selection for all a state’s electors (as it currently works in all but Maine and Nebraska), but a proportional voting system of some sort (so that, e.g., in Massachusetts you could still get Republican/Conservative electors, and in Texas you could still get Democrat/Liberal electors).
b) Electors are allowed to campaign on principles (e.g., pro-life, pro-gay marriage, deficit hawks, defense hawks, etc.), but they are not allowed to say “Pick me because I will vote for Y individual to be President.” This helps prevent the system from eroding into what it has basically become nowadays, which is a presidential-selection-by-proxy process.
2. The electors then gather, like the Catholic College of Cardinals, and deliberate behind closed doors as to who the next President should be. They can pick whoever they want – there is no process otherwise for limiting their choice (such as, e.g., a referendum selection process requiring them to choose from a “top 5” or what have you), thereby allowing them to consider the most qualified individuals rather than being forced to choose among politicians who happen to win popularity contests.
3. Once the electors have made a selection – say, by 2/3 majority, to force compromise choices instead of divisive options – they then propose that selection to the people.
4. The people then have a vote on that recommendation in a simple “yes”/”no” referendum. A majority of “yes” votes selects the President. A majority of “no” votes requires steps 2.-4. to be repeated. There will always be one or the other, and no one would ever be selected absent at least a bare majority of the people wanting them (as opposed to now, where a plurality vote-getter opposed by a majority could still become president).
5. If steps 2.-4. must be repeated, then:
a) After the first failed vote (or if the electors take too long in making a proposal – say, thirty days), the incumbent must leave office and a temporary caretaker must be installed until a new President is selected. The caretaker should be an individual or entity that could not possibly function in the office indefinitely (to avoid prolonged incumbencies and permanent impasses, raising the specter of autocracy) – e.g., Supreme Court, D.C. Circuit, a board of state Governors selected in a totally randomized process, etc. This would put pressure on the electors to make a selection, and it would put pressure on the people to approve a selection, as a dysfunctional executive branch would not be tolerable for too long (presumably, anyway).
b) After the fifth failed vote, the 2.-4. cycle ends and the five selected candidates are then put on a ballot, after which an instant runoff election commences to select one of the five to replace the incumbent caretaker individual/entity.
In this way, there would be democratic elements and an ultimate democratic check, but we bypass the most immediate direct-democracy problems presented by those “ill humors . . . [which] sometimes disseminate among the people” that the Framers feared (The Federalist No. 78). Electors would be accountable to their constituents, but only loosely so, so that they might be free to vote their conscience rather than what a faction among their constituents might prefer.