I go to SLC about once a month on business. Any place I've gone there that has a full bar will have table placards or price sheets on the table. If you consider using brand logos and signature cocktails for the featured spirits portrayed brightly on the placards as advertising, then yes, they allow advertising.
It's not that alcohol is secretive, it's that there are so many convoluted rules surrounding the alcohol. At the brewpub I often go to, you have to declare whether you are going to eat a meal or not, and they'll seat you accordingly. Diners sit in the 'restaurant,' which has a full bar, others sit in the 'tavern' where you are not required to order a meal, but you are only served beer and wine. I was told that technically their establishment is two businesses, with a shared kitchen, and seating separated by an aisle.