Interesting...the Republican and Democrat parties would do all that they could to get as many people in their party to sign up for this lottery so that whomever is elected would vote for their agenda. Then they would mold them into their fold as soon as they are elected with promises of goodies to come after they leave office.
But you would more likely have a few people that would get through with their own ideas, but they would be minimal. Like now.
I'm sure that would happen to a point; but the inner circle party members who hold most political offices today are a tiny minority (<.01%) of the total out there. I don't think they would be numerous enough to swamp the system on their own.The second-tier party faithful are far more numerous, but would prove more difficult to control, as their ties with leadership are looser and they have not been subjected to the same level of party "discipline." If they won office in this sort of system, they would owe no particular debt to their party for this achievement, and so a significant lever of influence is eliminated.
By contrast, the traditional electoral system is a closed shop. Either you or your desired candidate belongs to the inner circles of the ruling party structure (Republicrat in USA; LibLabCon in UK; CDU/SDP?FDP in Germany) or you are utterly out of the game, unless you are enormously wealthy.