All councillors should aim to ensure council proceedings are lawful, although the clerk is the best qualified to advise, or intervene if necessary. Although the chair is often the most visible councillor, they have no authority to make decisions and should only represent the council in respect of policies or decisions decided on by the council.
I'm not sure why a co-opted member would be without a vote. In any case, you are right that it is not the job of any councillor (including the chair) to police other councillors. As you say, anyone, including councillors, can make a complaint to the Monitoring Officer. So I would agree that it is the responsibility of each councillor to decide about interests.
I'm not sure that it is necessarily wrong for councillors to point out a known conflict, or to remark on the general principles governing personal interests. Although if this is done, it should be done tactfully. But none of them, including the chair, have any authority to rule on such matters.