To get back to the OP: What exactly is the purpose of supporting a “third party” candidate?
Like, what is that supposed to accomplish that electing a decent Democrat, or decent Republican couldn’t?
A “party” that just elects one person to one office isn’t a party. It is an independent candidacy, not a third party candidacy. For a real third party presidential candidate there’d have to be an actually existing third party. A charismatic billionaire celebrity that creates their own organization from scratch doesn’t make that organization a third party.
Take, for instance the Reform Party. For a few minutes it looked like the Reform Party might have turned into a real party. But it was always an organization to facilitate Ross Perot’s presidential ambitions, and once those were over the Reform Party disappeared.
But let’s not quibble about that. Third party/independent candidates for president don’t make much sense. Suppose you’re a charismatic billionaire celebrity who wants to become president. What’s the simplest way to accomplish this task? By winning the major party nomination of whichever party is most congenial to you, and then winning the general election as the standard-bearer of the new and improved major party. If you can’t even win the Democratic or Republican nomination, how the heck are you supposed to win in the general election?
Even if there exists a white knight who can ride into Washington and clean up the swamp, why does it help that this white knight is not a member of either major party? Why does it help that this white knight has no fellow party members in the legislature? The presidential elections are already about the particular candidates, the candidates define the agenda for the party. If the white knight cares about issues X, Y and Z that are neglected by both parties then they bring those issues up during the election. You think Karl Rove is sitting there waiting to cut the Republican candidate’s mike the second he starts talking about issue X?
The OP imagines everything backwards. That the problem is the fault of the major parties, and therefore we need someone not affiliated with either party. But the major parties consist of officeholders and operatives and voters. If the problem is that the voters keep electing Republicans, then how is a candidate on the ballot that’s not a Republican supposed to help? If the problem is that there’s this great person who could solve our problems waiting in the wings, except they can’t get elected because they don’t have a D or an R after their name, well, that doesn’t make sense because a person can join a party simply by saying “I’m a Democrat” or “I’m a Republican”. There’s nothing stopping a supposed “centrist but fiscal conservative but social liberal” from just running as a Democrat or a Republican, because guess what, we already have one of those, he’s the President right now and he’s a Democrat.
The problems our country has are too big to be solved by a white knight. In fact, no country’s problems can be solved by a white knight. And if you’re a charismatic billionaire celebrity who thinks you can solve the country’s problems by being a good person, and then you decide you can’t join one of the major parties and wrestle it into submission and turn it into the kind of party you want it to be, then you’re exactly the person who won’t be able to actually govern the country.
The president isn’t the national mascot. They have to appoint officeholders, sign or veto legislation, conduct foreign policy, command the military, and on and on. If a Democrat can’t appoint appropriate people to run the Defense department, or the FBI, or the State dept, and so on, what makes an independent president suddenly able to make the right decisions? That they aren’t hamstrung by their own party? Except being held back by your own party only matters if you care about not annoying them. And in actual fact the president is the leader of their party, not the other way around. A president’s appointments have to pass the Senate, if they can’t even get their own party onside to support their appointee, how would that be easier if they didn’t even have a party to get onside?
And on and on. When a major party president wants something done, they get their team in order, then sell it to the country, and either override the opposition or get the opposition to support the program. An independent president doesn’t have the advantage of a home-team who can be counted on to support them just because they’re a member of the home team. And if the Republicans automatically oppose the president just because he’s a Democrat, you think they’ll roll over on their bellies if he’s not? Or if the problem is that Republicans will vote against anything the Democrats propose, no matter how reasonable, just to be obstructionist, then how about we stop electing so many Republicans?
If the new president’s proposals are so popular with the country that the two major parties won’t dare to oppose them, how does this change just because the president himself is a member of one of the parties?
Or to put it a shorter way, if a white knight charismatic billionaire celebrity with a hankering to become president appears, and they decide not to win over one of the major parties to their side, then they’re an idiot have no chance of actually becoming president. Why couldn’t Ross Perot have run as a Democrat, or as a Republican? Because he had no chance of winning either primary? Why was that, if the country is clamoring for what he had to offer? Even if you’re just treating the presidential run as an extended media campaign in favor of your pet issues, why can’t you do that in the party primaries?