This is a very good posting. In my experience most people need to be asked! Perhaps it's about feeling appreciated and valued. Also some people mistakenly don't think they have the skills or time to take a leadership role. What may be worth doing is a quick skills, knowledge and experience audit to see who has different strengths. Then you can approach people with the suggestion that each family plays to their strengths. For example a family with an iPhone may organise a digital day outdoors. The family who likes football might organise an alternative footie session in the woods, etc.
Here in the NE of Scotland we have an interesting development. A grassroots outdoor learning group (ONE PLAN - Outdoors North East - Play, Learn And Network) has been formed which meets once every two months. We had our second meeting 2 weeks ago. In order to share the workload, the group decision has been to meet in different venues. The organisation that hosts the meeting will organise the activities and programme. Given the number of organisations and individuals who can do this, means that the workload is shared quite considerably ie no one person should have to do this more than once per year or even 18 months! Part of the reasoning behind this is to help people understand that everyone can share and it's about give and take.