I felt like that at times also. Although mine never did have tantrums like I've read about. But the climbing out of the cart, reaching for items on the shelves, etc. ? Oh yes.
And leave them at home? Umm.. not very practical. That would mean that I spent the entire time my husband was at work not accomplishing the things I needed to accomplish, and then when he got home and we were supposed to be having dinner or going to bed or spending "family" time, I was rushing off to run errands or do the "Walmart" hustle. Leaving husband at home irritated that the minute he got home from work he was "abandoned" with a kid jonesing to get out of the house and see the sights of Walmart.
It's just silly to expect moms to do that.
But I will tell you, that it does get better eventually. Gosh, I thought those days would never end... but they do. My son is 15 now, and he and his sister both went through a phase where when we went to the grocery store they preferred to sit on the bench up near the cash registers and play their DS game while I shopped for groceries. They knew that if an elderly person needed to sit down, to get up and offer the bench to them. They were very well behaved. (tween and early teen ages).
Now at 15, he goes with me and walks around, randomly asking if we can have "x" for dinner and asking for various fruit to try from the produce section. And calmly "suggesting" Little Debbie cakes when we go down the snack aisle. And debating with me the virtues of the Jalapeno Cheddar Cheez-Its over the Monterey Jack or Colby ones.
Drives me nuts b/c I spend more $ when he goes with me, but he is a joy. Keeps me smiling and laughing throughout the entire grocery shopping "ordeal".
Your day will come, too. All those years of having them underfoot and in (half in/ half out) the cart while you shopped will have dividends down the road. My son has watched me from an early age avoid buying the "impulse" items at the checkouts. And letting other people with only a handful of items skip in front of me in line. And making sure to put things back on the proper shelf when I change my mind about purchasing an item. And how to order deli meat at the deli counter. And how to evaluate the best bargain (sometimes the smaller containers ARE the better deal). And how to choose produce. And check egg cartons for damage before putting them in the cart.
And how to control himself. And how to have patience. And how to wait until we get home to open containers (we do not open food and eat it in the store--it MUST be paid for before we open or sample. That's my rule.).
But yes... the seemingly non-stop litany of corrections seems like it will be a friend forever. It won't. But when you can, try to offer up something for THEM as part of the ordeal. It helps give them something to look forward to, particularly on those days when you just KNOW that they aren't really up to it. A cookie from the bakery just before wait at the checkout line (the grocery stores usually give the kids one free). A few minutes on the toy aisle at Walmart/Target to look at, try out something they don't have at home. "Window shopping" for kids, if you will.
At Target, a bag of popcorn to nibble while you are shopping can be great, too.
Hang in there. You're doing it right. And it won't last forever.