with insects, cutting them up a bit will help the ants get the parts they want easier, and they may leave more of the discard where it is. And always with feeder insects be sure to give them a quick 5 seconds of boiling to kill off any parasites they might carry.
While i don't know details about many differetn ants. It is more common that as long as the ants don't have too much nest space, they will remove refuse from their nest to the outworld. But too much and they may use the trash to make their nest space smaller. Which is not good to have rotting refuse piles in among them like that.
In that case it is a little less about how much food you offered them and more about if they feel like their nest space is too big for them or not. Though as i read around it seems like some ants are kind of just filthy living, while others are much more fastidious.
For a protein source that's really clean i use fish flakes/fish food.
Specifically this brand:

Ingredients are listed like people food in order of % amounts largest first. A lot of the cheap/common fish flakes have a lot of flour/filler. These are mostly mysis shrimp and fish.
There is no trash it's 100% food to them. And very easy to give them frequent tiny amounts, rather than larger amounts that could become rot before they are used.