OP, we packed our lunches for our 9 day trip last week and had no trouble finding a place to sit and have our picnic lunch. Each day we packed sandwiches, fresh fruit, yogurt, chips, string cheese, snacks, and frozen juice boxes/water bottles. It was stored in our backpack cooler, which was stowed away in the bottom of the stroller until time to eat. The security guards at the baggage checks were envious of our lunches!
In the MK, we ate lunch one day at Pecos Bill (not crowded, plenty of tables), and another day we ate dinner at Tomorrowland Terrace (we were the only people on the level where we were sitting). Tomorrowland Terrace was actually better, though it was outside, as it was cool under the fans and the kids got to run around without us worrying about them bothering other guests.
At AK, we ate at a table that was not attached to a restaurant or anything. It was hot, outdoors, middle of the day, crowded, but we were fine.
Epcot: twice we had lunch in the WS - once in the China pavilion (in the almost-empty quick service restaurant at a table) and on the brick retaining wall at American Adventure.
We never got ugly looks or comments from CM's or guests as we enjoyed our picnic lunches. We decided to go with picnic lunches for a couple of reasons. The first is it is way less expensive! While we could easily afford to eat chicken fingers and cheeseburgers, it didn't sound appetizing for a whole week for us. Also, the food we packed was far more healthful than what we would have been tempted to purchase. We also wanted to be able to eat when we were ready, or rather, when the kids were ready, and didn't want to try to find something my picky little ones would eat.
All in all, over 9 days, we ate out just 3 times: once, on my date night with my husband while my mom watched the kids (Rose & Crown, followed by dessert in France); Pinocchio's Village Haus; and once we got Chinese food in China while it was raining and we were stuck for a little while. (We cooked dinner each night in our condo. Actually we used our crockpot for dinner 4 nights, the other nights we made tacos, etc.)
Bottom line: do what is best for your family! It works for many people. When my husband and I go for F&W in the fall, we will probably not pack lunches. But when our 3 kids and my mom are with me, packing lunch works best. In my many years of packing lunches, I have never been given a hard time by CMs or security or even other guests.