Multiplying two univariate Gaussian PDFs

$$ X \sim \mathcal{N}(\mu_X,\sigma_X) \\ Y \sim \mathcal{N}(\mu_Y,\sigma_Y) \\ Z = XY $$

results in closed form equations for $\mu_Z$ and $\sigma_Z^2$:

$$ \mu_Z = \frac{\sigma^2_X*\mu_Y+\sigma^2_Y*\mu_X}{\sigma^2_X+\sigma^2_Y} $$ and $$ \sigma_Z^2 = \frac{\sigma_X^2*\sigma_Y^2}{\sigma_X^2+\sigma_Y^2} $$

some derivations of this here

This is a direct consequence of the exponential functions and the addition of exponents.

But, it is said by a lot on here that it is "obvious" that the multiplication of two Gaussian random variables is not a Gaussian random variable.

I just don't get why it's obvious. There seems to be a move to jump to the integral to combine two random variables. Why? Why is multiplying two PDFs not the same as multiplying two random variables?

My motivation: there is a literature within psychophysics to use the combinations of gaussian functions to approximate sensory cue combination in humans. An example. I want to extrapolate this to a similar case, but I'm running into this theoretical distinction people are making that I don't understand.

Links to other posts about similar things that don't quite answer this question:

Is the product of two Gaussian random variables also a Gaussian?

Texbook with proof that product of two Gaussian functions is also Gaussian

What's the densitiy of the product of two independent Gaussian random variables?

Product of two Gaussian PDFs is a Gaussain PDF, but Produt of two Gaussan Variables is not Gaussian