Every chore system starts the same way: a grid, names across the top, tasks down the side. It survives about three weeks. Someone is away, someone is sick, someone has finals, and the grid is now a description of a household that no longer exists.
Fairness over a month, not a Tuesday
The useful question is not "whose turn is it" but "who has done the least of this lately, and is actually here". Rotation should route around absence rather than pretending it away, and it should even out over a month instead of demanding a negotiation every time life happens.
That also removes the argument. When the rotation is visible and the history is visible, "I always take out the trash" is a claim anyone can check, and it is usually either true and fixable or false and settled.
Done is a fact, not a promise
A chore marked done by the person who did it, at the time they did it, is worth more than a chart someone updates on Sunday from memory. Small point, large difference: it is the difference between a record and a reconstruction.