Are there any tournaments, set up similar to brackets, that allow the 16 entrants to play all four rounds, with the overall lowest scores taking first, second, third, and fourth places?
Have a look at this one: The WGT King's Cup
It's not a bracket, but it's 4 rounds of single-play, with the overall lowest scores taking first, second, third, and fourth places.
WGT publishes the weekly WGT King's Cup tournament each Thursday, so look for it then. EDIT: Here's a link to this week's tournament.
All Bracket tournaments will be exactly as you describe, with luck-of-the-draw playing a part. That's the definition of a Bracket tournament.