In some other nations, there are political positions of trust where when people take them, they traditionally (and depending on the nation, may be legally required to) renounce their party affiliation while in office (and sometimes for the rest of their life). For some of these nations, this is no doubt primarily symbolic. For others, there is additional leverage that comes from belonging to a party; parties may expel members for voting the wrong way or doing the wrong things; leverage seen as inappropriate to be held over these positions.
The Supreme Court Judiciary in the United States is the closest we have to this concept (as far as I'm aware); I don't believe the justices need to unregister with their party, but they are expected to refrain from political statements, acts, and generally donations.
Would it be helpful for US Presidents (and candidates) to have a similar tradition when elected and to have candidates be merely "endorsed" by parties? Would reelection be more difficult? What are the values, political, and logistical issues at stake here?