Post-quantum alternative to ElGamal? (public key verifiability) Are there any alternatives to ElGamal that would be resistant/annoying to quantum machines?

I would like to preserve public key derivability/verifiability -- ability to ensure existence of corresponding private keys for made-up public-keys.

I've looked up several other cryptosystems from NIST proposals of post-quantum cryptography, but by directly reading the papers it looks like they lack this property.

For instance NTRUencrypt is a lattice-based cryptosystem. 
As it operates under a lattice field in theory public keys could be randomly generated without private ones.
But I'm not sure that there would be a way to ensure existence of private keys and possibility of decryption (although the private key space could be large enough and this assumption would somehow hold).
There is nothing quite meeting your requirements, especially if you require the ability to generate random valid public keys without knowing the corresponding private key. CSIDH (not a NIST candidate) comes closest, in that you can at least generate random private keys and easily publish the corresponding public keys verifiably.