Inspiration from work kinda makes me want to introduce a STRESS penalty rule. Basically responsibilities and worries accumulate, these distractions represents how much levels of mastery in skills are there to compensate for this. That higher levels of skill is not just for greater feats of ability but being able to overcome great amounts of distractions (basically representing greater and greater levels of "keeping it together" and concentration).
Leaders who are responsible are the first to take a lot of these penalties. Maintaining discipline and example is STRESSFUL as well as pay extra attention. Ad-Hoc a simple "Stress Penalty" would suffice for every responsibility. Delegation or Strategic re-organization of Assets and Resources would be the typical solution to this. The rule should be compatible and work well with Multi-Tasking rules or acting as such for systems with no such rules. Characters who are not "organized" cannot see all the tasks they need to do, they can be aware of only a few tasks.
I think this kind of rule makes players appreciate the burden of multiple responsibilities and juggling these. It also incentives them to seek strategic solutions to minimize this and taking too many responsibilities on. It also simulates how mess the PC's life can get.