If CAs have to share CTs and have to save everything the CT would save to their last submission then no CA can destroy the log without colluding with other CAs.
(I.e. your log ends abruptly but polling any other CA that published to the same CT shows there is more including reasons to shut you down.)
I don't see how a scheme where the CT signer has this responsibility makes any sense. If they stop operating because they are sick of it, all the CAs involved have a somewhat suspicious looking CT history on things already issued that has to be explained instead of having always had the responsibility to provide the history up to anything they have signed whether or not some CT goes away.
(I.e. your log ends abruptly but polling any other CA that published to the same CT shows there is more including reasons to shut you down.)
I don't see how a scheme where the CT signer has this responsibility makes any sense. If they stop operating because they are sick of it, all the CAs involved have a somewhat suspicious looking CT history on things already issued that has to be explained instead of having always had the responsibility to provide the history up to anything they have signed whether or not some CT goes away.