IMHO, the "strong case" refers only to the way the proposal needs to be presented. There is no reference, in the original sentence about the "importance" of the establishment of the club. (设立辩论俱乐部的重要性* ) We don't know whether the people presenting the case of setting up the club will stress on the importance of its establishment or on some other reasons.
The idea of the sentence is that "we better make sure we make a very convencing case to the people who are going to decide on this matter." So on top of the 力陈 , perhaps we should add something like the following.
To make people feel this proposed club is of some importance, right? In vigorously reasoning out (力陈) the importance of such a proposed debating club, one will have to present his arguments, won't he? Therefore I see little contradiction between the two versions which, imho, are just two ways of expressing the same sentiment. Thanks for your alternative version!
原文：Arguments can always be made aobut th esignificance of one date or another, but politically a strong case can be made that English society began to take on its 21th century guise, began to change in character, during the year 1910.