4 candidates go on the general ballot.
Every voter can vote 1 vote for any of them.
The top 2 go to the second round of the general ballot.
Minimal or no vote splitting.

How do we get 4?
One way is the 4 that get the most signatures, or the 4 that pay the highest fees. I prefer the 4 who get the most people to donate $5 each.
This way, anyone can run, unless the race is already very competitive. There should be no minimum number.

Since there are 2 general races, we drop the public primaries to save time and money. Instead, we let parties and independent voters hold their own private convention.
Then, we don't let parties just put someone on the ballot. All candidates must compete to be in the 4 that get the most signatures or most $5 donors, whatever the standard is.
If someone can't get into the top 4 on their own, they drop out and team up with a similar candidate.

I have other simple fraud prevention if you are worried about that. How do you like the basic idea?