"Vote for 1, Top 2 Win"
In the short term (0-1 years), we'd see districts elect either two standard Republicans, two standard Democrats, or one of each.
In the medium term (2-7 years), I think we'd see districts get a choice between a conservative, a moderate, & a progressive, and elect 2 of the three.
In a longer term (8+ years), I think we'd get a real third party that stands for different things than either of the Republicans or Democrats. The political spectrum would change into a political triangle, each party having a base in one corner & competing to win over the voters in the center and on the sides between each other.
A governing coalition in a three-party system is just 2 of the 3 parties. So nearly 2/3 of voters are represented in the governing coalition.
If you increase the number of the parties, the number of people represented by the governing coalition goes down! e.g. With any even number of roughly equal parties, you only need a bare majority of members. With odd numbers of roughly equal parties, you need 3/5, 4/7, 5/9, ... trending closer and closer to a bare majority as the number increases.
So given roughly equal parties, a 3-party system maximizes the proportion of people represented by the governing coalition.
With a governing coalition of about 2/3rds of the seats, the coalition can lose nearly 1 in 4 of its members and still move forward. Consequently, it becomes very safe for individual members of Congress to stick to their principles. So we would expect to see representatives show much more backbone, express much more individuality, and more closely follow the local culture & interests they represent rather than a party line.
The output of the coalition needn't be bland compromise either! With lower pressure on every members to conform, they can afford to put political capital into unconventional ideas that they know won't win over every member of the coalition -- as long as it's an idea that has at least some appeal across party lines.
The opposition party has no hope of stopping the majority through gridlock, nor of winning an outright majority. Instead, their only strategic route forward is to work on compromise bills with one side or another of the majority, in hopes of later winning them over to a coalition.
3-way polarization is not a risk. All the incentives are against it.
What about excessive depolarization? Would voters start to see "not a dime's worth of difference" among all three parties? Political theorists in the mid-20th century thought we needed more polarization to give voters a more meaningful choice. (How short-sighted those theorists look now!)
The answer is No, we wouldn't have excessive depolarization -- because we have primary elections. They didn't have those in the mid-20th century. They only had the general election, which is dominated by moderates. Primary elections are dominated by ideologues. We'd keep the benefits of primaries in generating meaningfully different ideas, and yet not suffer polarization over it.
How might "Vote for 1, Top 2 Win" fail? That's always important to consider.
It might fail if regions are so politically divided that they can't unite around just 3 parties. With 4+ parties, the advantages largely disappear. (This risk is greatly reduced by making the Presidency a 3-way race.)
It might fail if a large region is so united that it elects a single party. (This risk is greatly reduced by laws ensuring competitive elections.)
It might fail if a coalition between two parties becomes so habitual or even formalized that voters treat it as a single party. (This risk is greatly reduced by regulating the parties to forbid excessive entanglements.)
Those are the main predictable failure modes, and none seem especially risky or severe. So I say let's go for it!