There are a lot of things they could have wanted to change about the contract besides just a pay cut. They could have wanted to change the length, they could have wanted to remove the no trade clause, they could have wanted to add or remove guaranteed income or bonuses, they could have wanted to add or remove incentives.
People are mistakenly assuming the only thing the Saints could have wanted was a paycut. They could have wanted several other things. Maybe they did want him to take a paycut, but we don't know that. People have just assumed that's what it was.