All of these cap maneuvers are accounting games that we in the business call "timing differences." The bottom line is if you pay a player $X, you will eventually have to apply every cent of that $X against in your cap. The question is just when it will happen. And as I've seen mentioned in this thread, you can only push that application out so far past a player's expiration date. And if you try to balance this out by playing the same game with newer players you're just delaying the inevitable.
I would guess that the Saints probably had some kind of 5-10 year plan (post-Brees) to gradually take their medicine, hoping to remain competitive during that period. But then COVID resulted in a down cap year which screwed up the plan. And then the GM just couldn't himself on signing player Z. And now they're four years into implementing the plan and it's still ugly. They've had mediocrity for a few seasons, but it's like they have to start over with a new 5-10 year plan to clean up the cap gradually. Either that or maybe, as some have suggested, you just take your lumps and knowingly punt on a season or two.