Why can't Sanders, Clinton, Trump, Cruz, Rubio, etc. just run as independent choices where the winner is chosen from overall vote count?

Sanders and Trump were neither Democrat nor Republican, yet they joined for the ability to be elected. If those two ran independently against Clinton and Rubio/Cruz, it would be a good race where all four could debate and get a fair share of the votes.

Why is it not organized in that manner, why must it be Group A or Group B?