Why in RSA do we use mod n rather than mod phi(p⋅q)?
There are two reasons for not using $\varphi(n)$ during encryption.
The first one is that it doesn't work - you can verify this by looking at the RSA correctness proof. It requires, that the modulus is $n$.
The second one is - if you know $\varphi(n)$ you can efficiently compute $d$ given $e$, and therefore know the private key.