would this three party system work?

[Mister Mets]

Due to the way the primary system is set up, certain groups will have a disproportionate influence on the major parties.

If you go from two major parties to three, their influence on an individual party can be even stronger since they're going to be a larger share of the primary vote. The alternative is that there may be a schism of some sorts, IE- if some members of the religious right stick to the AIP, while others go to the universalists.

African-Americans would probably have a larger influence on the American Labor Party than on the Democrats.

We're not really accustomed to three-party elections in the US and the ways a strong third party candidate can help the greater of two evils win. You could look at the 2010 British parliamentary elections where conservatives got 47% of seats with 36% of the vote, and Liberal Democrats only got 9% of seats with 23% of the vote.

We're also not used to it being the norm for politicians to not have a majority of the vote. That can make it difficult to govern. Obama's difficulties would be even greater if he won 40% in a three way race.