What is the purpose of calculating the N modulus in the encryption phase of asymmetric cryptography?
What am I missing?


Not much; you've just come across one of the reasons why we never use "raw RSA".  Instead, whenever we use RSA (say, for encryption), we always pad out the message so that, after padding, it's not much smaller than $N$.  This prevents the message from being less than $N^{1/e}$ (which, as you point out, would be completely insecure), and it also prevents subtler attacks based on the homomorphic properties of raw RSA (namely, $RSA(x)RSA(y) = RSA(xy)$