This allows majorities to more honestly differentiate between their preferred candidates and also support consensus candidates without the fear that they will end up with no utility at all. So in situations where a majority is split between two candidates, they can show their honest preference for either without letting the minority beat both on total points.
The two situations where this rule breaks down are when 1) voters cast protest votes for unviables, raising the bar for a majority and 2) a minority honestly more intensely prefers its candidates than the majority. The first could perhaps be solved by having a dynamic plurality requirement i.e. 40% instead of >50%, and may not even be a big deal to begin with for viability and quality, and the second is probably not a very likely or harmful situation overall, and may even make regular Score less viable than this.
This idea probably doesn’t have much room to grow with Score and STAR already out there, but perhaps it could be floated around for legislative decision-making or private elections, as a way of balancing utilitarianism with respect for the majority’s wishes.
One alteration is to instead eliminate candidates who don’t receive >50% support (more than a 5/10) from a majority; this could be called the “candidates require a majority of support from a majority” version. And a further alteration, to address situations where a minority more intensely prefers its candidates, is to only trigger the rule when a majority gives at least one candidate >50% support, but these complicate the method too much to be viable.