Is there a public-key cryptography method that enables public keys to be much smaller than private keys? Does a public key cryptography method that enables the public key to be orders of magnitude smaller than the private key exist?
Yes. Hash based signature schemes are a prime example. The private/secret key of a Merkle tree is the whole list of leaves, while the public key is the root hash.

In practice the leaves are not usually all stored, but derived from a shorter key. Even then it may be useful to store e.g. $O(h)$ hashes to accelerate signing, where $h$ is the tree height. With CMSS (pdf) $T$ levels of trees are generated and the signer stores the verification keys for the active tree at each level, which is about $T$ times as much data as the public key.