One way to get students to do things that you want them to do and are valuable when they do them is to actually require it. Simple concept.
Since the papers are supposed to be read in advance, but few people do, perhaps, you want to use a bit of stick to goad them on.
One simple solution to both problems is as follows. Suppose that there are five papers to be reviewed in a session. One individual is responsible for presenting each of the five, of course. But require that all the students, prior to the session submit at least one question to you directly on at least three of the five papers.
It might be too much to require questions on all the papers, so I suggest only some, rather than all.
You can use the submitted questions in various ways. You could ask them yourself during the session, naming the originator or not. You could ask the originator to ask their question. You could also present those questions to the speaker a bit before the session.
Getting the questions directly and in advance gives you an opportunity to comment back on the quality of the questions. This could be either privately or in a session in which you generally discuss the sorts of questions that are most valuable to ask about papers in your field.
One possible flaw is that some of the papers don't get covered with the above scheme. In that case you could assign, say, one of the three papers to each student. They then need to provide at least one question for that paper and at least one for the remaining two.
I've assumed that someone is in charge of the session, such as a professor. If that is not the case, one participant can be chosen to "lead" each session and manage the questions. Leadership can rotate. And for a volunteer group, make this part of the charter that people agree to when they join.
I don't know how this would scale. I've assumed a pretty small group. But I think such clubs are going to be small scale in any case.