You're right. It's too strong a claim to say that individual consumption has zero effect on the economics. The better claim is that an individual choosing to stop consuming something has no material effect on child labor. It has a material effect on a specific quantity of product purchased in specific location during a specific time period. That quantity is not directly tied to the amount of child labor consumed. The economic system has layers upon layers of buffers between the consumer and the child laborer and the individual choices would need to build to point where they can have a material impact on the first buffer before they can have an impact anywhere else.
This is why the actual effective boycotts are about demonstrating power, not about actually causing a mass quantitative change through changes in consumption. The ability to do it deliberately and in a focused way demonstrates effective organizing that could be used at a mass scale, and thus the capitalists are required to meet the demands of the boycotters in order to prevent them from actually wielding that power to do mass economic damage.
It's NOT the individual choices that make this change, though individual choice has a role to play as discussed elsewhere. It's the ability to organize and demonstrate the power of the organization that makes this change. It has nothing to do with the actual loses of profit and everything to do with the risk of future losses of profit.