Responsibility gets handed out on the basis of evidence, and the evidence is whatever you have already been given. Nobody can tell from a conversation whether you will finish something; they can tell from the last thing you finished. Small tasks are not a holding pattern before the real work. They are the interview.

Look at the smallest thing currently assigned to you and do it completely, including the part nobody checks. Hand it back early rather than late. Then wait, because that is usually how the next thing arrives.