Randomness in Public Key Encryption
A public-key encryption algorithm cannot rely exclusively on "public" randomness.  The reason comes down to the fact that encryption is a public algorithm that can be run by anyone. If the algorithm's output included all the random coins that it used, then an adversary could determine the message from the ciphertext, by running the encryption algorithm with the (given) coins and each of the two candidate messages from the IND-CPA experiment. The true message (and only that message) would yield the target ciphertext.

In some cryptosystems, the decryption algorithm recovers all of the encryption randomness. Examples include encryption schemes based on injective trapdoor functions like RSA. There's no contradiction with security here because one needs the secret key to recover the randomness.

For your second question, we can't go so far as to say that all publicly recoverable randomness is non-security influencing, i.e., replaceable by a fixed public constant (which is the contrapositive of your question).  For example, consider a PRG that uses some random public parameter along with a secret random seed (e.g., an LWE-based PRG).  Now consider an encryption algorithm that chooses both a public parameter and a seed for the generator, to produce a pseudorandom string for the "real" encryption, and outputs the parameter as part of the ciphertext. This would be secure, but might not be if we fixed the public parameter to be, say, all zeros.