Not sure how you pull it off, but I like the general concept.
It would take four NATIONWIDE primary elections to determine who advances to the final two candidates. Voters would be asked to go to the voting booth 3 times (and a 4th time in the General Election)
Sweet 16 National Primary Election, held first Tuesday of April preceding the General Election: Sixteen candidates on the ballot, the top 8 advance
Elite Eight National Primary Election, held first Tuesday of May preceding the General Election: Eight candidates on the ballot, the top 4 advance
Final Four National Primary Election, held first Tuesday of June preceding the General Election: Four candidates on the ballot, the top two advance.
The final two candidates would be brought to the national party convention for selection of the party candidate to face the other party’s candidate in the General election.