Aldi's is known to purchase similar meat and produce lots as
Walmart, which are inferior to the likes of what Publix, Kroger, etc. purchase. It could very well be that your local Publix doesn't properly care for its meat and produce, diminishing its quality and shelf life. As I mentioned earlier, these are the same lots that the likes of Whole Foods and premium grocers purchase -- the only difference is handling. Whole Foods spends more on labor than the likes of Kroger, etc., which enables the chain to better handle produce, maintaining its quality and prolonging its shelf life. Their are premium grocers that go beyond Whole Foods -- e.g. putting out a limited number of each item at a given time, having an in-store banana ripening room, etc. This is obviously labor intensive and requires more frequently deliveries, which inflates the price. It's been a decade since I've been into a Publix, and while they never reached the handling level of Whole Foods (let alone the premium grocers), they were known for handling it better than most chains (e.g. Kroger). Maybe that's changed.
And while I'll maintain that the average shopper could save significantly more by shopping sales and pairing with coupons at a traditional supermarket -- this requires both stockpiling and preparing meals around sales (and again, if you don't have room to stockpile or need a fixed weekly grocery budget, this method won't work for you), Publix would likely be an exception. Publix sells the idea of a high level of service, and it has the highest margins of any large traditional grocery chain.
FWIW, for many years we would do all of our shopping at discount chains like Aldi (when we were traveling), Walmart and 99 Cents Only. When we switched to shopping at traditional supermarkets, we realized how much longer our meat and produce lasted, which helped us reduce our shopping trips and ultimately saved us even more money.