Why does this not work: Alt. way to compute the period of a discrete periodic signal that is a sum of 2 complex exponentials
The flaw in your reasoning is that you only consider the cosine term. The cosine term has a period of $48$. But there's also the complex exponential, which is not just a magnitude but a function of $n$. You can show that that exponential term is periodic with a period of $48$. However, you cannot conclude that the multiplication of the two terms must also have a period of $48$.

Multiplying the two terms results in a period of $24$ because

$$e^{j\frac{8.5\pi}{12}n}=-e^{j\frac{8.5\pi}{12}(n+24)}$$

and

$$\cos\left(\frac{0.5\pi n}{12}\right)=-\cos\left(\frac{0.5\pi (n+24)}{12}\right)$$

so the sign changes cancel in the product.