Sorry if this is a duplicate question, but I haven't been able to find this exact question. Is the product of symmetric positive semidefinite matrices positive definite? is close, but for reasons I point out in a comment below, I don't think it provides a clear answer to the questions.
Edit 1: Update! I better understood Robert Israel's answer at the other post. Let me rephrase my question. I have a "proof" below that shows the product of two real, symmetric, positive definite matrices $A, B$ is also positive definite. But this is apparently false. Can someone help me find the error.
Consider real symmetric positive definite matrices $A, B$. This is defined as $A = A^T, B=B^T$ and $\forall x \neq 0, x^T A x >0$ and $\forall x \neq 0, x^T B x > 0$. Choose an eigenvalue-eigenvector pair $\lambda, v$ of the product $AB$:
$$ AB v = \lambda v$$
Left-multiply both sides by $v^T B^T$.
$$ v^T B^T A B v = \lambda v^T B^T v = \lambda v^T B v$$
Since $B$ is positive definite, we know that $v^T B v > 0$. We also know that $B^T A B$ is positive definite. Rearranging, we see that
$$\lambda = \frac{v^T B^T A B v}{v^T B v}$$
Since both the numerator and denominator are positive, $\lambda$ must also be positive. Since this holds for all eigenvalues, the product $AB$ must also be positive definite.
I would hope somebody would correct me if I'm wrong, but I see no mistake in your proof. It is clearly laid out and each step follows logically. In fact, the exchange that Dietrich Burde linked in the comment seems to back up your assertion as well.