In a time when partisan rancor is at unprecedented highs, and the Congress is incapable of passing an act to name a post office, why should the President be beholden to one party or the other?
Sure, parties are probably a smart way to organize the Congress, in an attempt to herd cats, but the Supreme Court at least puts up a pretense of non-partisanship. And not being affiliated with a party gives the President the ability to work with either party, or both.
Parties have no place in the Constitution, they're not mentioned at all.
Given my druthers, I'd amend the Constitution thusly:
1. The President is elected by direct nationwide popular election. All candidates are listed by name only, no party affiliation. The rules for the Presidential election are set by the Federal Election Commission, not by the states.
2. On the first Tuesday of November, we have an election (plus early voting, absentee voting, all that good stuff).
3. If nobody gets a majority, on the third Tuesday of November, we have a run-off among the candidates who received 10% or more of the vote.
4. If nobody gets a majority, on the fourth Tuesday of November we have a run-off between the top two from the second round.