I don't think an Aldi or Trader Joes is a Publix threat...totally different stores. We've had both in Atlanta for along time with no affect on Publix or Kroger. Both are like a mini-Costco would be for those people that really have a hard on for Kirkland Brand. Aldi and TJs are basically the same thing, tailored to how you want to think about yourself. Neither is worth a damn for meat or produce.
We hit Aldi about once a month for a handful of items for which their brand is as good a name brand for considerably cheaper...cereal, chips/snacks, salad dressing, nuts, cookies, cheese. They're cheaper than even the Publix/Kroger store brands, and in some cases better. Now, I wouldn't make a special trip to Aldi to save 75 cents on a bag of chips, but when you have three teenagers, and are going through five boxes of cereal, three bags of chips, 20 yogurts, etc a week, on a monthly basis it adds up to being well worth the trip. You've got to work your way through the products though and figure out which ones work...some I like as much or more than a name brand...other things are definitely no good.
Whole foods is stupid expensive, but I could see myself shopping there when it's just my wife and I eventually...an extra few bucks isn't a big deal on a small total, but it's a total non-starter for the full family. Sprouts isn't that expensive on a lot of things, but the selection is still not good enough to be a one-stop for a big family. Been to Earthfare a couple times and didn't see anything to recommend it really. They're a worse whole foods, not having the selection of really good bakery/deli/etc that Whole Foods has. My favorite of the fancy shmancy places is probably Fresh Market, but they're too expensive on the lower priced things that we go through a lot of. That's probably the one I would choose to shop at though.
Kroger and Publix, we have to bounce back and forth, but they've got the price/selection/quality/cleanliness combo in the Atlanta market, nothing competes, and I don't see anything competing unless Wegmans hits town.