Is there any advantage using this kind of ROM-based randomized hashing in digital signatures?
You are essentially describing a salted hash which is very important for some post-quantum signature schemes. In some lattice-based schemes it is important that the same signing key never be used with the same hash so that some message-replay defence is required. Schemes such as FALCON achieve the anti-replay by salting the message before hashing. Your method is one way in which this could be achieved.