I don't have any historical foundation for a state electing house members simply based upon a statewide election as opposed to districting but I do believe that it did exist early in American history. Obviously there's been a problem with gerrymandering which is inherent when politicans are the ones creating Congressional districts. This problem is actually becoming worse today because analysis of voting is much easier with computers.
As for why eliminating Congressional districts would increase the likelihood of third party candidates it's a matter of simple logic. If we have ten Congressional seats and three parties we assume ten members of both the Republican and Democratic party to be running. Each party wants all of the seats. There is going to be one favorite Democrat and one favorite Republican and perhaps only one third party candidate. These three candidates are going to receive the vast majority of the votes from their party members. The favorite third party candidate is probably going to receive more votes than the lessor members of the D's & R's because they will receive almost unanimous support from their party members. The third party candidate's vote isn't going to be split and will almost always exceed the vote for the lessor D's & R's. This would bring diversity to the House which is obviously missing today.
Once agian the key is that the US Constitution leaves the matter of voting for House members to the State legislature (so long as it's a popular vote) just as it leaves the matter of selecting Electorial College members to the State legislature (and no popular vote is required).
There is not now nor has there ever been a Constitutional requirement for a popular vote for the president. In one sense I'd like to see the popular vote for president eliminated completely. It's turned into a partisan popularity contest where the voters are generally unininformed related to the candidates.
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