(Of course, sometimes those future savings wouldn't have happened anyways, due to unreasonably optimistic assumptions about the performance of the economy.)
Second, even if everybody knew who was going to be the loser from the changes 8 years in advance, that's not enough. That party would have to be strong enough under the current system to undo the changes. In practice, that would require controlling the House, the Senate, and the Presidency... unless both parties decide that they would lose power to third parties, and strike a deal.
E.g., in the first cycle, extreme candidate A wins the actual election while the moderate, B, is shown to be widely tolerated under the approval system. Nearly half the voters--those who voted for the opposite extreme, C, end up loathing the results of the actual election and wishing in retrospect that the approval system had been in place. Perhaps in the second election cycle, candidate A wins re-election because they're a known-quantity/incumbent.
Finally, in the third cycle, history starts repeating itself and people get tired of A's party. E.g., the Overton window slides back towards C's perspectives leading to C being seated. This time the people who still prefer A are disgusted with the outcome and prefer the outcome of the approval system.
At this point, half the voters have long argued that the approval system makes more sense and the other half has also come around to thinking maybe it could bolster their position as well. It'd only remain for less partisan activists to e.g., get a petition going towards a referendum.
It's hard to predict the short-term beneficiary parties when it goes into effect, sure, but its quite easy to predict the long term ideological benefit of increases in proportionality on modern democracies.
Ironically, that may make near immediate implementation more viable, if their is a divergence between the expected short-term and long-term impacts, and those expected to benefit believe they can use that to preempt the naturally expected long term benefit but those who would expect that long term benefit don't believe their opponents will succeed at that.