I can say with near certainty that this is not the case. We have a few customers who open carry at my store (in the heart of Midtown Memphis, lol) and one of my other managers questioned our policies on that as she didn't like it. She was told from on high to leave it alone as it is legal here and that we can to only ask them to cover the weapon if they receive a complaint from another customer. Walgreens isn't a gun friendly company (employees banned from having weapons on property) but it is more concerned with alienating customers. Things may be a different in Nashville, or more likely is that manager that OP spoke to was an idiot who was talking out of his ass.
This is the most likely reason the sign just recently went up. Walgreens started selling beer in all stores just prior to the laws regarding carry changing. At that time we received an email with PDF files of the signs we needed to print out and hang. That sign was included in the group of signs to post. Some store managers were a little slower than others in posting all of the required signage. Nothing was ever sent out saying that sign was no longer required after the law change. It is likely that there was a store visit, and the store manager was told that he was missing some of his/her alcohol signs, and all of a sudden, that sign shows up along with the signs regarding underage sales and pregnant women drinking, etc.