Sorry, still a circular argument. You have no more "given away" anything than the toilet manufacturer does when he sells a restaurant a toilet. Both were compensated for the product they produced. Heck, a toilet maker would have a better claim that one "has created an atmosphere where the establishment wouldn't profit as much in abscence of public use of a product" since no one goes out and buys a certain brand toilet because they used it in a restaurant - like people do with music they would have otherwise never heard.
The musician has *not* been compensated. There are different rates for putlic and private performance licenses. The club (if it paid for a CD at all) paid for a private license, then used it for a public performance to enhance its own profits.
It's not up to you to decide what I charge for my business. If I think a loaf of bread is only worth a quarter, I can't walk into a store, throw a quarter down and walk out. If I think it's too pricey, I can just do without, but I can't walk around paying what I think something should be worth and just taking off with the goods.