Primaries and the Conventions

In a system with political parties, you need candidates who are aligned with parties. Parties choose their candidates who, if not members, are at least generally in agreement with the party, will join their legislative group if elected to a legislative position, and will generally vote to advance the interests of the people who comprise the party. 

Parties do this in a wide variety of means around the world. 

Some can be literally appointed by their leader, which happens in Canada in some instances, particularly in the case of an early election caused by the collapse of legislative confidence in a parliamentary system, or in areas where the party is weak or their local branches are corrupt. 

Some are chosen by a panel which acts for the purpose of considering candidates. New Zealand Labour uses this method using a few different selection groups to hold basically a job interview, advisory votes by the members, and the panel makes the final selection for each constituency, and for the New Zealand mixed member proportional system, a committee basically comprised of their party's executive board (elected at annual conventions with delegates of all the branches and affiliates voting) and a few other people take the people chosen to be candidates for local positions and puts them in a rank order. 

Some parties do this but with added steps. NZ Greens do much of what Laboru does but their members hold a confirmation vote of the choice of the committee, which can force them to go back to the drawing board. 

Others have delegate conventions elected by lower party meetings. Germany uses this system to have each of their 299 constituencies each have delegates elected by the local basically precinct or municipal branches vote to elect each local candidate, and then state conventions with delegates elected by the constituency conventions forming the full list for the state's list ranking, and all votes are held by secret ballot when choosing individuals for positions or list rankings all the way down until the members directly elect delegates at the lowest level; this chooses the candidates for elections of the German Bundestag. 

Some countries have moved to more open processes, although there is debate about whether this exact idea is the best way. In France, some parties like the Socialist Party hold a vote where all people in France who are members of the party or are 18 years as of the date of the general election, who pay a nominal fee of one euro and subscribe to a basic set of principals, could vote. America makes primaries largely based on that although without the nominal fee and without needing to subscribe to values, just declare a party in the case of closed primaries (and register by the necessary deadlines. It would be possible to make it so that a person couldn't vote in two party's primaries both meant to nominate candidates for the same position. 

In a system that is organized most like America's but with proportional representation in a multi party system, the most likely way (and one of the easier) is to simply have all parties vote by single transferable vote (which is party agnostic and also works for internal elections without any organized parties or factions in turn) among those who vote within them their slate of candidates, as many winning (and the quota set based on) how many seats are to be contested in the general election for the district, so if the general election has 4 members to be elected to the district, each party nominates four people by STV. 

This means that each party will nominate a pretty diverse slate, likely representative of the major factions represented within. In a 4 member district on these rules, any faction with more than 20% of the votes can be represented.

Bear in mind that in a proportional system, more parties are also likely, and each party will solidify around their core support with those significantly disagreeing able to easily form new parties (or run as independents) without much consequence. Ergo, there wouldn't be much of a Bernie-Biden split in the Democrats, you'd probably see at least one party for the centre Democrats, and at least one wing for progressives, and maybe one for the more socialist-green wings, and Republicans would split too. Third parties may do so as well, such as if the Reform Party (which Trump interestingly ran in back in 1999) survived, you'd see splits in their factions too, or at least the codification and consensus of the party would make it easier to understand what those parties stood for. And so you wouldn't be choosing so much from these factions within one party but factions on more technical differences and about the personal qualities of the candidates. 

The general public would then be able to choose for themselves which party they primarily support (and end up being willing to transfer votes to if necessary), and would choose from among the factions. Maybe a political party only has weak support for a faction within it but they could be quite popular with the general electorate. Maybe an above the line option like Australian Senate elections allow would occur, where if you choose it, and there are four seats open, then 1/4 of a vote is assigned to each candidate within the party (or if they didn't get enough candidates, maybe just a third of a vote to three candidates). 

The factions within parties doesn't work much with presidential elections. There is only one winner of a presidential election. But even still, you would be ranking your choices in a direct presidential primary like this. You would likely need support outside your core base and would easily end up with five or more strong candidates at the time of the voting. They need to unify the party across their factions to win, and to drag along with them as many wins at lower levels as they can. The same is true for gubernatorial candidates and mayoral candidates. 

Some elections in the United States are technically non partisan, like many local elections as well as school boards and special district elections, most judicial elections outside of the Deep South, and generally officers related to law enforcement like sheriffs and district attorneys, one Nebraska Senate, and a few others. Primaries become redundant, and anyone who wants to run would simply file a petition with signatures and maybe a fee. Nobody in these races needs to appeal to the factions within a party to win, and they don't need to court that many votes as if they were trying to appeal to small differences within a party, they know that you rank candidates in order of favourite to least favourite, and all the voting will be done by those at the general election.

Another issue is that most people don't remember this, but primaries often elect those who serve on party committees, the most well known of which being the Democratic National Committee and the Republican National Committee, those who serve as delegates to conventions, at least in some manner, such as the direct election of precinct chairs and committees or conventions which elect in turn those above them. Single transferable voting for both types and ranked ballots for their chairs and executive officers like treasurer would be a major change, which provides for representation of the minority viewpoint within the party and within any delegation (so geography matters less) and with parties being larger in number and in a wide range of options for people, on a bigger spectrum, parties would be keenly aware that they can't just be the opposite of the other main party but would have to keep their voters from going elsewhere with attractive policy. Their leaders would generally have to deal with more factions to be elected and would have to appeal to them as second and third choices who might be removed if they can't keep everyone aligned. 

These committees and conventions also elect other bodies like the Democratic platform committee which will propose the Democratic platform to the Democratic National Convention in a couple weeks. It just voted down Medicare for all (although it's not particularly necessary to be specific about medicare for all as opposed to say the Scottish NHS or Dutch multi payer insurer system that are still universal healthcare systems either way), but the wisdom of doing so or taking other stances on political questions, or not taking stances, would be critically important when trying to create a platform people align with. Poeple's concerns over the personality of a person is mitigated with having many candidates from the same party to vote for, many parties to pick from, and that no single party is likely to have enough support to win the big things like the presidency, laws, budgets, taxes, and appointees on their own, so policy can be more important at this point. In the progressive primary for instance, you probably wouldn't see the titanic fights that Bernie Sanders put up with the relatively centrist faction, you would just see a relatively boring fight between a couple of progressives from the outside. The influence of the formal committees and their leaders may also be more limited due to the need of candidates to appeal more to those outside the party, the diverse slate of candidates that may represent those not of the party central dogma, and with any bias less influential over what the aggregate sum of a person their candidates are. 

These committees also know that outdated policies that they don't defend well and without the perception of a conflict of interest such as the perception of superdelegates among the Democrats could cause them to be replaced, so staying on top and purging the worst of your policies and old legacies you don't believe in any more is helpful. It might even cause parties to completely sink, probably removing the old associations of say the Democrats and the KKK or the Republicans aligned with Herbert Hoover, just as nobody really associated Robert Borden with the modern Conservative Party in Canada, and so potentially limits prejudice from that angle. 

What the total impact of this generally would be is that politics doesn't feel stuck or back and forth ad Sisyphus.