Well it's only the third time War is happening, so things are still being worked out, naturally, with the system. It will be a long while yet before two Wars in a row have the same ruleset; and it will indicate that we've become experienced enough to know exactly how best to run it.
Right now one of the weaknesses in War is with the masters system. It is far too easy for a master to be overturned in the trials system ... and really the problems that you have described, SG, those of lack of leadership in War, stem from inactive or poor mastery. Putting a better premium on leadership -- by making it harder to become or replace a master, or making it more closely related to reliable activity rather than a burst of activity; or by changing the way War leadership is determined (i.e. lieutenants are chosen in a different process) can contribute to this. Still, having all kinds of roles and responsibilities in War just doesn't make sense the way it's done now, as a backup plan or otherwise.
If an entire team is of inactive players, it doesn't matter what system you have anyway; they will be inactive in any case. If there is a team with a few inactive leaders, it's actually better to use a system with fewer roles so the one or two more active players can handle everything. What you're talking about is essentially having someone to blame for anything that doesn't get done so that a penalty is connected to a name.
The best solution may be this: have major penalties of inactivity threaten a master's position. Every time a master's team takes a card penalty related to inactivity, that master will receive a -1 penalty in the next trials. Combined with making it difficult to overturn a master, this will guarantee over the long run that only active and responsible players are masters.