I think there are some problems with the system you're setting up.
You would very quickly have de facto parties. If parties are outlawed, and candidate John Smith promises to vote for Paul Ryan as Speaker, we would conclude that he is the equivalent of a Republican.
There could be some advantages, with more shifting alliances than we currently have, when there's less of a perception of partisanship with certain actions. It would be a bigger deal for a Senator to go from the Republican party to the Democratic party than to change the individual who they're backing.
An issue with top two jungle primaries is that it can allow for crazies to win. In the French presidential elections, one of the two final slots has often gone to a white nationalist party. In Louisiana in 1993, the final two in the gubernatorial race ended up being Edwin Edwards, a former Governor who had been indicted, and David Duke.