Questions about proof of correct encryption in the Paillier cryptosystem
Revealing $r$ would then allow the verifier to prove to someone else (another verifier) that $c$ encodes $i$. The verifier could also prove other things knowing $r$ to a different verifier (any other proof using a paillier ciphertext, the corresponding plaintext, and the random value $r$).

With the ZKP, the verifier cannot prove anything to anyone else about that ciphertext. This seems like a useful property in some situations.

To answer your second question, assuming what I have said is the only side effect of revealing $r$, then your method would also allow the verifier to prove things about the sum to another verifier. That may or may not be an issue with your system. There could be other side effects that I am not aware of.