It's not just about it costing more. It's in part that if there were a large tax, there wouldn't be as big of a difference between a $13 ice cream and a $16 ice cream as there was between a $3 ice cream and a $6 ice cream, so the better product would do comparatively better in the market. Therefore a better product would be comparatively more common. It's also that I would be discouraged from binging on $3 ice cream, making what times I do indulge a more rare and luxurious experience. That second part I can do myself; I just don't buy as much ice cream and when I do, I buy it better.
At the extremely high end of luxury goods, a certain class of product (Veblen Goods) is actually more desirable based if sold at a higher price. But that doesn't mean people buy equivalent goods priced lower and ask to pay more. The stated sale price itself has an impact on customer satisfaction because it implies the seller's belief that the product is higher quality, and in some situations the higher price simply makes the good more "exclusive" which appeals to certain (snobby) buyers.
Isn't economics weird?