Not true, we do it all the time and manage good control of our dining expenses.
All Disney resorts have fridges and in mine is ham, turkey, sliced cheese, mustard plus breakfast items and a few drinks. We have bags of rolls and rectangle container to put sandwiches in. Pringles don't crush in a backpack.
We eat breakfast in the room. Yogurt, bagel, creme cheese, bananas, juice pouches.
Pack lunch for the parks. Sandwiches, pringles, bottles of water but will occasionally buy a bottle of soda.
We put the food in a backpack with cold bottles of water and it's good until lunch. There are plenty of spots to eat.
Depending on park and hours we might eat TS or CS dinner, go to resort or eat in room if not very hungry.
You can bring your own food or have it delivered to resort if flying in.