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The voting for small parties problem could also be address with multiple ranked choice voting.
That way, candidates who don't win, can have their voters submit second and third choices, that are count when or if their primary choices don't win.
That way, people can vote for their actual favorite, while without taking their vote away from a second favorite with better chances at winning a seat.
This is especially important for positions like president, where only one candidate CAN win (some countries achieve the same with multiple rounds of voting, but you can theoretically achieve the same representation in one round).
Ranked choice voting is more relevant for elections where one winner must take all (for instance presidential elections or voting districts with only one representative) which will always suffer from not being able to represent everyone
Direct proportional representation gets around that problem by avoiding situations where one person has to "win" an election
As for the president problem, you can get around that by simply not having a president with any meaningful power. The prime minister can be appointed by the cabinet, and the head of state (be it king or president) can be a purely ceremonial role.