You can fire people and/or let them go without claiming you "take full responsibility". That's ok, and exactly my point.
The phrase "I take responsibility" is never accompanied with any sign that the person does in fact take responsibility. It means more than just "Ooops, I did a bad"
Also if one of your options for "taking responsibility" is quitting, I guess your view of the world is that you would also fire any of your employees when they mess up, when whole industries have recognized the need for blameless cultures where making mistakes is part of the learning process.