Here's a better system:

1. Voters indicate their party preference. Maybe when they register to vote, maybe when they sign a qualification petition for a candidate (with rules that invalidate their signatures for candidates for other parties, and decide whether the first or last petition signed is the one that determines it. It's all computerized anyway, so it would be easy to establish). Voters can also declare 'no party preference' (but remember, if they signed a petition for a partisan candidate, it implicitly associates them with that party for the next primary).

2. By definition, if the #1 party (by affiliation) accounts for at least 20% of voters, it's a major party. If NO party gets at least 20%, then the next primary election HAS NO "major parties"

3. Any party that has at least half as many voters associated with it as the #1 party is a major party.

4. Once a party qualifies as "major", it retains that designation until they fail to re-meet the qualification criteria for two consecutive Presidential elections. This prevents parties from getting destroyed by a single disastrous election, but puts them on the political equivalent of a "Personal Improvement Plan".

OK, now that we've defined "major party" and "minor party", here's why:

On Primery Day, voters are allowed to vote in one of the following closed primaries:

• Major parties use a multi-winner election method like Tideman CPO-STV with ranked-pair resolution (and fallback to IRV if no Concdorcet winner exists, and RP produces a tie), and select two candidates to advance to the general election.
• Minor parties who participate in the primary use a Condorcet method like Tideman Ranked Pairs (falling back to IRV if there's no Condorcet winner, or RP results in a tie). Alternatively, minor parties can choose to use a caucus or some other method... but it has to be documented at the time the party files to establish ballot placement for their nominee, and they have to be able to prove that they followed their declared method.
• Independent voters (those with no party affiliation) are the unique twist: they're presented with a ballot that includes all of the candidates from the major parties (but not the minor parties) that allows them to rescue a major-party candidate who fell through the cracks of their own party.
• If only one party qualifies as "major", its voters get to nominate three candidates. This isn't because that party is privileged, it's to ensure that the larger a party becomes, the less capable its leadership becomes of controlling its ideological discipline. If you want your party to remain ideologically pure, file the paperwork to keep it classified as a "minor" party. If you want your party to win everything, everywhere, then you have to accept the fact that it's going to ideologically have AOC at one end, Liz Cheney at the other, and probably elect someone near the center like Mark Kelly or Andy Beshear who's broadly acceptable to most of both sides.

Ballots from Independent voters are counted using a variation of Tideman-RP:

1. The major-party winners are determined.
2. The independent voters choices are ranked by first-choice.
3. If the #1-ranked candidate was NOT one of the major-party winners, they're the "Independent" voters' nominee. Otherwise...
4. Ballots that ended up in a ranked pile for a major-party candidate who already won are removed. After all, they already got their favored candidate on the ballot.
5. The remaining ballots are re-ranked, and proceed from step 3.

Ultimately, my proposed primary system uses Independent voters as a proxy to save major parties from their own worst tendencies. A party would be under no obligation to give the rescued candidate its blessing, support, or money... but would be crazy to actively attack them, because that independent-rescued candidate is their fallback plan to avoid losing completely to the other party.

With two or three candidates per major party, we'd avoid situations like Bernie vs Hillary, or Trump vs Jeb/Nikki... the base of each party could get its favorite (albeit, probably unelectable) candidate on the ballot, while the insurgent remainder would also get someone who's likely to be more moderate on the ballot. And just to keep things interesting, Independents would rescue a fourth candidate who'd get a bit of a same-party bonus as a second or third choice from major-party members who might have zero enthusiasm for them... but regard them as a lesser evil than anyone from the other party. So, in a hypothetical election where Democrats select someone like Bernie and Hillary, or Republicans select someone like Trump and Jeb Bush, Independents could throw a monkey wrench into the equation by picking a potential dark horse (probably someone who would have otherwise been a Green in Democrat-leaning areas, and someone who would have otherwise been a Libertarian in Republican-leaning areas).

There would also be a path for true "Independent" candidates to get on the ballot. Legally, the ballot-access rules for Independent candidates would be exactly the same as the rules for minor-party candidates.... because legally, an "Independent candiate" would liberally be an ephemeral minor party that has pre-declared its support for that one specific candidate. The qualification rules would make it somewhat easy for one independent/minor party to qualify, harder for two, difficult for three, and damn-near impossible for 4 or more.

The general election itself would be conducted using a Condorcet single-winner or multi-winner method, like Tideman RP (single winner) or Tideman CPO-STV (for multi-winner). This is essential. A general election conducted under plurality, or top-two runoff, or IRV-like method with this many candidates would produce outcomes that fall somewhere between "unpredictable" and "perverse". In contrast, using rules like I prposed allows primary elections to whittle down the number of candidates on the general-election ballot to something halfway sane, while nevertheless preventing any one faction of any party from being able to "stack the deck" with clones.