Why do exponents not distribute over addition?

Multiplication distributes over addition [..], so if [exponentiation] is just repeated multiplication why shouldn't the same be true [..] ?

Multiplication distributes over addition, and if you repeatedly multiply by the same quantity or quantities, then that too distributes over addition: for example, $c(a+b) = ca + cb$ (doing multiplication once), $c^2(a+b) = c^2a + c^2b$ (multiplying by $c$ twice), $c^n(a + b) = c^n a + c^n b$ (multiplying $n$ times by by $c$).

But repeatedly multiplying $a$ with itself, repeatedly multiplying $b$ with itself, and finally adding the two is a different thing from repeatedly multiplying by $a + b$. There is no reason to expect that repeatedly multiplying by different things should somehow distribute over repeatedly multiplying by something else.

The quantity $(a + b)^n$ means that you repeatedly multiply it by $a + b$. The quantity $a^n$ means that you repeatedly multiply by $a$, and $b^n$ means that you repeatedly multiply by $b$. As you're multiplying by different things, the results can be arbitrarily different.

In the specific case of $(a + b)^2$ for instance, we can see that it is $(a + b) (a+b)$, and as a multiplication, it does distribute: $(a + b)(a + b) = (a+b)a + (a+b)b$. But $(a)(a) + (b)(b)$ is an entirely unrelated quantity, so it doesn't make sense to expect them to be equal.

Perhaps this image answers your question: