Your general frustration at feeding a bunch of people with varying preferences mirrors my frustration. I feel like all i do is make food and clean up from it. It is exhausting.
I think a 600$ budget for basically 3 adults (teenagers probably eat even more than adults) and 2 kids is reasonable. I wouldn't kill myself to get it lower unless you really need the savings.
For summer one thing I am doing is making an all-day snack container for each kid. I put a variety of stuff, about a days worth, and they aren't allowed to ask me for a snack until they've eaten the stuff I set out. Today it is peanuts, fruit, cucumber slices, a few crackers, and a few chocolate chips.
Gardening saves us a lot if you are into that.
For breakfast I often make a crockpot full of oatmeal the night before. Easy, and I make 2 days worth so I only have to wash the crock every couple days. Everyone adds their own toppings in the morning.
For dinner the disparate tastes are really a pain. I typically make what I am going to make, do it as customizable as is reasonable (ie tacos are great because everyone adds what they want) but in the end, eat it or make yourself a peanut butter sandwich. Don't like it? Fine, you'll live until breakfast. But I'm not a short order cook, I'm not making you something else.
Off topic, but perhaps of interest to people reading this thread: My mom is a diabetes educator and obviously the high obesity rates among children, and the huge uptick in children with diabetes are of great impact and concern to her. One thing they are working on educating people about is that you shouldn't give kids a snack after a meal if they didn't eat their meal. The snack food Americans typically consume is far less healthy and I would argue more expensive, than the meal food.
I have the best luck with meal planning and shopping if I stick with a regular rotation of meals so it can be mostly on autopilot, with minor variations to account for what is on sale. So we pretty much always grill once a week in summer (exactly what we grill I can decide on the fly at the store), have pasta and tomato sauce (buy a bunch of pasta whenever it is on sale, same with tomato sauce, its always one hand), mexican-ish once a week (taco salads when lettuce is crazy in the garden, taco/rice bowls, or in taco shells/tortillas if those are on sale) etc. That way I can basically shop with no list since its the same general stuff week to week. I try to also look up new/different recipes for a few meals a week so we don't get bored, but even if I fail to do that (often!) I have 4-5 basic/easy/cheap meals I can shop for off the top of my head, and then add in fruit, veggies, milk, eggs, etc.
ETA: for school lunches, we almost always pack the night before. I tend to buy one special convenience thing for the week (yogurt tubes, crackers, etc) and that is it. The rest is fruits, veggies, etc. We do buy bread sometimes which some people might consider convenience since you CAN make it at home pretty easily. If you and your H do it together at night would he be more open to trying your way? One thought would be to make something the kids like, such as crock pot mac and cheese, and divide it into a bunch of lunch-sized portions on Sunday, so all your H has to do is grab one for each kid. Same for cutting up carrot sticks, washing a bunch of fruit, etc. For canned soup alternatives, again make a big batch in the crockpot or stove, and divide into a bunch of portions in the fridge or freezer.