the most voted for parties

Simple 'first past the post' systems like they have in the US are flawed. The biggest problem is that clones (candidates or parties with similar positions) split the vote. For example, suppose 10% of the population wants Evil Dictator, but the other other 90% each want one of 18 different candidates as their first preference, evenly divided on first preferences (so 5% on first preferences), but rank any of the other 17 higher than Evil Dictator. So Evil Dictator has 10% of first preferences, but is the last preference for 90% of the population. The other candidates have 5% each.

First Past the Post would elect Evil Dictator in this circumstance. Better electoral systems (e.g. the Schulze method) would elect one of the other candidates.

This applies still if you elect a plurality of people - e.g. there could be two Evil Dictators, who 90% of the public oppose, but who have the highest vote because there are fewer of them to split the vote. Better systems like the better STV variants ensure proportionality (it avoids a landslide where the same voters determine all the representatives in a winner takes all approach). A larger parliament means more representation of the perspective of smaller minorities - so they are at least heard.

A "score" based voting system, if it is just a ranking of parties, could work like this. But if you are suggesting adding up the votes (so, for example, a 5 is worth 5x as much as a 1), the problem is tactical voting. People will, in practice, vote to make their vote count them most.

Let's say, for example, there are three candidates, Racist Evil Dictator, Racist, and Progressive. Let's say we know for granted almost everyone is going to score Racist Evil Dictator as 1. If a progressive was voting honestly, they might vote Progressive as a 5, and apart from the racism Racist might have been doing well, so they'd get a 3. The racist supporters, however, if they were being honest, would give Racist a 5 and Progressives a 3. Let's say there are 1000 progressive voters, and 600 racist voters. If voting honestly, the scores would be Progressive = 5 * 1000 + 3 * 600 = 6800, Racist = 5 * 600 + 3 * 1000 = 6000, Racist Dictator = 1 * 1600 = 1600. Now the problem is, you can't really get people to vote honestly. So let's say Racist riles up their followers to instead vote Progressive as a 1 to, even if they don't really think that. Now the scores are Progressive = 5 * 1000 + 1 * 600 = 5600, Racist = 5 * 600 + 3 * 1000 = 6000. Racist wins.

In practice, when a system allows people to vote tactically and have an advantage, it becomes a race to the bottom. That's how you end up with dynamics like the two-party system. A good voting system works by removing incentives to vote tactically - if you put your true preferences down, you will not be disadvantaged in your influence on the election, even if other people attempt to vote tactically. That means that genuine third parties have a chance if the people like them, even in the absence of coordination.