Does there exist a proof-of-retrievability scheme that is publicly-verifiable, limited-use, and does not use homomorphic encryption?
I do not know of any approaches in context of proofs or retrievability (PoRs)/provable data possession (PDP) that use homomorphic encryption. However, many of those schemes employ homomorphic (linear) authenticators/tags for the metadata such that the proofs delivered by the server can be of constant size, i.e., by aggregating single tags.

Now to some misunderstanding regarding the type of verifiability: 

Public verifiability in this context means that the data owner who initially outsources the file is the only entity that can modify the data, but a third party, which is given some public information, can only verify a proof from the server that the data is available, i.e., check that the data is still available/retrievable. 

Consequently, public verifiability always involves some secret that is only known to the data holder (who creates the file and metadata and may modify the stored data and the corresponding tags) and some public information (corresponding to that secret) that can be given to some third party such that it can trigger proofs for availability of the data (but cannot modify/replace the data). Thus, the party in possession of the public information can only check if data is available, but can not modify already stored data in such a way that a proof after modification of the data will still succeed.

As a consequence, MAC based PoRs/PDP schemes, which are essentially the same but have different security models, can never achieve public verifiability as it is defined in this setting, as it would require to share the respective secret (and this allow modifications). The same holds for any secret key only scheme.

Clearly, the use of a privately or publicly verifiable scheme to allow another third entity (distinct form the data holder) to request and verify proofs from the server depend on your trust model. Public verifiability assumes that the third entity is semi trusted, i.e., only trusted to conduct the proofs honestly, but you do not trust the third party that it wont modify the stored data. If you fully trust the third party, i.e., you are sure that it will not modify your data unauthorized, you can also use a privately verifiable scheme and simply give away your secret to the third party. However, this is a quite strong assumption.

Typically, such PoR/PDP schemes are either private verifiable or public verifiable. But there are also protocols which achieve unbounded private and public verifiability at a quite low cost simultaneously (based on the same metadata/tags). You may look here for such a scheme. However, it is based on pairing friendly elliptic curves and may not be that straightforward to understand as you say your knowledge is quite limited.

A very simple and easy to understand privately verifiable but bounded use scheme is presented here, but it does not support public verifiability as discussed above. Nevertheless, it is easy to implement and may be sufficient for your purpose.