We control the costs through where we shop, how often we shop, and minimizing waste.
We do the bulk of our shopping at Sam's Club (family of four), with one-offs purchased from the "fancy" grocery store. I am in the military, and we have access to the on-base Commissary, which is basically a non-profit grocery store. I find Sam's prices to be comparable (I price checked ~90% of what we buy on a regular basis and it was equal or cheaper, almost without exception). The produce and meats also tend to be better at Sam's. Costco would be another option if you have it.
Wife and I share a shopping list on our phones for each store and align purchases with already scheduled trips -- almost never do we just drive to the store to get something. If we're planning X for dinner and realize we forgot to grab something we generally just roll that meal to another day vs. going to the store (unless spoilage is a factor).
Rather than taking a menu approach, we pretty much buy the same stuff from week to week. It consists of mostly staples -- tomatoes, lettuce/spinach, cucumbers, potatoes, bananas, berries, eggs, milk, plain frozen broccoli, plain frozen chicken breasts, ground beef, some good cheese, etc. What we cook on any given night is a function of mood and what needs to be consumed. Tomatoes getting soft? Roast 'em. Bananas turning brown? Time for muffins/bread. Spinach getting mushy? Saute the whole container with Olive Oil and a little salt/garlic.
Not sure how you're preparing your food, but another cost savings is making the upfront investment in seasonings (vs. packets/boxed mixes) and learning to make sauces/salsa from scratch (vs. the bottled stuff). Moreover, you can dial everything in to the way you and your family like it.
When it comes to our kids (4 & 5) they eat what we eat. No exceptions. We set that expectation early (as we were raised the same way). There were a few nights of going to bed hungry / eating last night's dinner for breakfast early on. But the early pain was worth the gain. Kids will reliably eat damn near anything now (olives are a particular favorite).