Homomorphic encryption - Why does addition not imply multiplication? As far as I know:

There are some partially homomorphic encryption (PHE) systems that support either addition or multiplication.

A fully homomorphic encryption (FHE) system can do addition as well as multiplication and thus supports arbitrary computation on ciphertexts.

My question is (disregarding computational efficiency):

Why does a PHE-system that allows addition on ciphertext not directly imply that it also can do multiplication, since

$$a \times b$$

is the same as

$$\underbrace{a + a + \cdots + a}_{b\text{ times}}?$$

Are there some computations that are only possible with a direct multiplication instead of a continuous addition?
There are at least two problems;


The $b$-times addition leaks the $b$. A semi-honest observer can see that you add the $a$ by $b$ times. However, in FHE, the $b$ is also encrypted with semantically secure that leaks no information. The only information available to the observer is the circuit.
In FHE, the $b$ is coming (or may come) from another result, which means that $b$ is also encrypted. In additive PHE, you cannot multiply by $b$ without decryption.


You can look at some example of FHE circuits from this answer to see that some of them are not even possible with additive PHE.