Hash and (MAC) based signatures using zk-SNARK
This is a valid paradigm for building a signature scheme, although a secure commitment scheme should be used to commit to $k$ instead of just using $H(k)$ as the public key. This type of construction was published by Bellare and Goldwasser at CRYPTO'89; see New Paradigms for Digital Signatures and Message Authentication Based on Non-Interactive Zero Knowledge Proofs. Formally, they use a NIZK, as you stated.