If you read the instructions on the Insinkerator, it says you can't put grease, fat, or large food particles inside. It's basically for small stuff that got stuck on plates. It can gunk up the inside of the GD itself. Use cold water to flush it out, and sometimes grind ice in it to clean the blades.
If you don't want to dump stuff from the fridge into your garbage, you can find a spot in your yard, dig a hole, and dump the food stuff (no plastics) there. An elderly friend of mine used to dump eggshells and food stuff in a certain spot near his garden.
Some houses used to have grease traps for their kitchen sinks. I don't know if that is really a thing anymore.
The food stuff is why I miss having pigs-- they used to be the garbage disposal units for our house.