Yep - that is car camping. Driving into a campsite, really.
Most sites (almost all sites you pay for, not always the free sites) will have a grate over the fire-pit that you can use for cooking. Few have grills where you can use charcoal briquettes - more likely found in state and city parks. When we found one that didn't have a grill or fire pit, it was time to break out the PB&J sandwiches.
Your plan sounds good. Basically, unless you want
to eat junk, there are plenty of non-perishable food items in the grocery stores. You just have to make a stop every few days to keep yourself stocked in the perishable goods. We also kept nuts, dried fruit and granola bars on hand for the those moments when we really couldn't eat another sandwich, and just wanted a quick snack.
If you have a cast-iron skillet or dutch oven you can cook over a fire-pit just like you would over a stove top. We also had a tea-kettle/water pot to boil water, and if you have boiling water you can make a wide array of "quick convenience foods" - pasta, oatmeal, instant mashed potatoes. We also ate a lot of canned goods - soups, chili, stews, vegetables. Campfire spaghetti? No problem. Scrambled eggs for breakfast? Totally do-able. You just have to watch whatever you're cooking more closely because there is lesser heat control (things tend to cook faster over a roaring fire).
When we had a cooler with ice we found it lasted around two days before melting. Kept things cold for an additional day in the cold melty ice-water. We stored eggs, mayo, cheese, lunch meats and yogurt in there (just don't store anything packaged in cardboard - it gets wet).
The one item that we found very difficult to keep was ice-cream. I know, who tries storing ice-cream overnight on a camping trip? But we did, and it didn't work. Other than that, though, there are very few restrictions with car-camping since you don't have to worry about weight. Don't waste your time spending money on "camp food" - dehydrated meals that you can rehydrate with water. Those are good for hiking because they're light weight, but a complete waste of money otherwise.