The publishers can't entirely do the storage themselves since the whole point of CT is that they can't retract anything. If they did their own storage, they could rollback any change. Even if the log forms a verification chain, they could do a rollback shortly after issuing a certificate without arousing too much suspicion.
Maybe there is an acceptable way to shift long-term storage to CAs while using CT verifiers only for short term storage? E.g. they keep track of their last 30 days of signatures for a CA, which can then get cross-verified by other verifiers in that timeframe.
The storage requirements don't seem that bad though and it might not be worth any reduced redundancy and increased complexity for a different storage scheme. E.g. what keeps me from doing this is the >1Gbps and >1 pager requirements.