If I deliberately generated a public key that was divisible by 3

This is already well known:

When encrypting with low encryption exponents (e.g., e = 3) and small values of the m, (i.e., m < n1/e) the result of me is strictly less than the modulus n. In this case, ciphertexts can be easily decrypted by taking the eth root of the ciphertext over the integers.

https://en.wikipedia.org/wiki/RSA_(cryptosystem)#Attacks_aga...