Why are the concentrations in the equilibrium constant multiplied?
As discussed in comments, you can find a good derivation in a standard university level physical chemistry or thermodynamics textbook.

Let's instead consider why it should be true.

From a microscopic perspective, what we're trying to balance, in essence, is the density of states on both sides of the equation, in essence, the count of ways to rearrange the various pieces. These are heavily dependent on the relative spacing of energy levels and degrees of freedom on both sides. Fortunately, this complex information is captured in the value of $K$.

But the point is that if you're trying to find the total number of permutations of two different collections, you multiple the values. For example, if there are two dice, the total number of outcomes is 36.

And concentrations essentially measure the number of things to rearrange, so the total number of possibilities should scale with the product, not the sum.

This is all a bit hand-wavy, but again, I'm not trying to derive the answer. I'm just trying to suggest that even without a formal derivation, it seems very reasonable that the correct operation to use is multiplication.

EDIT:

After reading this again, I realized I can extend this with another hand-wavy justification.

Remember that we have:

$$\Delta G = -RT \ln K$$

If you have two reactions that you combine together, you want the energies to sum. But that translates into products inside the log! Once again, the proper operation for equilibrium constants is multiplication.