Extra fees of any kind just really, really annoy people. This is true at restaurants but also plenty of other industries. In the majority of cases, the loss of business from patrons who avoid your establishment due to the extra fees outweighs the revenue generated by the fee. Of course, you are welcome to test out your theories, but you're unlikely to find many people who predict they will be successful.
The only real question here is if it's to your advantage to advertise a lower price and then tack on a credit card processing fee or advertise a higher price and then provide a discount for cash/bank transfers.
The first gives you a lower advertised price but then irks customers when they find out about the extra charge. The second gives you a higher advertised price, which could be worse depending on the nature of the business.
But the proposed alternative where you charge the same to everybody and incorporate it into the advertised price requires you to use the higher advertised price, which has no advantage over the second option regardless of whether or not the second option is better than the first.
Restaurants, which is what your post is about, rarely advertise specific prices. Part of that is because a restaurant bill is comprised of multiple components -- entree, appetizer, dessert, beverage, "extras". Another part is that menus typically have dozens of options, another is that prices change often, and another is that many restaurants are local businesses with limited or no advertising budget. And a huge one is that restaurants that do promote low prices are often considered low quality as well.
In other words your point is pretty much moot because nobody is choosing a restaurant based on, veal parmigiana costs $25 at Villa Roma and $24.50 at Sicily Palace. But they very much might remember that Sicily Palace charges a credit card fee and boycott it in favor of somewhere else.
I'd like to believe this is true but in my metropolitan area, it's quite common to see e.g. a covid recovery fee, inflation recovery fee, iniative 82 service fee, and autograt when you are checking out. It's extremely bad for price transparency but the practice seems very popular among merchants who are betting that customers will still show up as long as the menu prices appear low.
Overall, I eat out less, but "drink out" more and never get more than one drink at the establishment. I can't claim to be able to uniquely isolate any of my behavior to fees because a lot has changed in my life. Before covid I was a student and now an FTE, I was shy and lonely and now I have an active social life, I lived in a very big city and now live in a big city, but I can tell you I get an ick about going out, and when I'm proposing to go out I am mindful of costs and fees and propose places that I consider to be cheaper and more forthright about pricing.
