Alternative Bracket for Regionals

I’ve never understood why D1 four team regionals are set up as they are where the winners of the first day’s game play each other on the second day. This “Present Bracket” guarantees that for the remainder of the regional one team will have a pitching advantage.
Why instead isn’t it set up that on the second day, the Winner of Game 1 plays the Loser of Game 2 and the Winner of Game 2 plays the Loser of Game 1. We’ll call this the “Alternative Bracket”. With this arrangement there are three possible outcomes:
A) Both Winning Teams from Day 1 also win on Day 2
B) Both Losing Teams from Day 1 win on Day 2
C) One Winning Team from Day 1 wins on Day 2 while the other Winning Team from Day 1 loses on Day 2.
If outcome “A” occurs, then two teams are eliminated and you’re left with the two remaining teams playing a three game series against the other on equal pitching.
If outcome “B” occurs, then all four teams are still in contention. In this event the teams would each play the team they’ve not played yet in an elimination game. The two teams surviving the elimination game would then meet in a single game to determine the champion. Once again, pitching is equal throughout the tournament.
Lastly, if outcome “C” occurs, then you’re effectively left with the precise same situation you have after the second day now, where one team is 2-0, two teams are 1-1, and one team is eliminated. In this event, the team that’s 2-0 will enjoy a pitching advantage.
The Present Bracket” system always results in a team having a pitching advantage after Day 2. Using the “Alternative Bracket”, it’s possible that one team will emerge with a pitching advantage after Day 2, but also very possible that teams will compete on even pitching though the entirety of the tournament.
Can anyone explain why the “Present Bracket” is used instead of the “Alternative Bracket”?