That comment is a misinterpretation of the mathematical definition of IIA - but it does raise a good point. The proposed system would, in actuality, be rather poor, it would encourage strategic voting to a tremendous degree, which would make it almost exactly like approval voting - which doesn't fit all criteria.
The US Congress is trying to resolve the national debt by getting hundreds of people to agree on a solution. This is silly. They should agree on the rules of a game to play that will result in a solution, and then play the game.
Here is an example game. Suppose there are N representatives, all with an equal vote. They need to reduce the budget by $D.
What game-theoretic problems does this game have? Can you think of a better game? Is it politically better to call it a "decision process" than a game?
The main trouble area, to my mind, is order of play. First I said that budget items would be listed by taking turns. The 1..N, N..1 order is supposed to make neither first nor last position preferable. But taking turns introduces complications, of not wanting to reveal your intentions early.
Then I said votes are placed secretly and revealed all at once. This solves problems about game-theoretically trying to conceal information or bluff your opponent. It introduces other problems, such as tragedy-of-the-commons scenarios, where every Republican spends their "defend" votes on some pork in their state instead of on preventing tax cuts, because they assume some other Republican will do that.
Is it better to play "cut" votes first, reveal them, and then play "defend" votes?
Is there a meta-game to use to build such games?