Wrong.
There is still consumer surplus, even if GDP is not increased.* This is formally equivalent to saying there is no consumer surplus sells his home to another private individual—after all, after the transaction, there’s no increase in the housing stock. But plainly there must be a surplus due to trade for both parties as a result of a voluntary transaction: sellers only sell when they value the purchase dollars more highly than the item conveyed, and buyers only buy when they value the item conveyed more highly than the purchase dollars.
*As it happens, when calculating GDP for a given year, economists do not count the purchase prices of new-construction homes in that year. Instead, they estimate “imputed rent,” or what the equivlaent in rent would be for all occupied homes in that year. Investment securities are not counted in consumption spending, so this doesn’t directly apply. What it does show, however, is that merely increasing the number of items to be bought and sold on the market is not the only way to increase the economy’s productivity (as measured by GDP). And so it is incorrect to say, because this activity (secondary trading) does not create a new item to be bought and sold on the market, it therefore has zero economic benefit for society as a whole.
Your “simplifying assumption,” which goes astray on this point, is actually an assumption that secondary trading is, by definition, unhelpful to the macroeconomy in terms of productivity. But as shown above, there is no reason to say that creation of new tradeable items is a necessary predicate to increased productivity. Given your assumption, it is no surprise that you are able to derive your desired conclusion from it. However, as it illicitly assumes the proposition it purports to prove, your syllogism has no probative force.