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The problem with crewing is that only the skipper/dad gets the experience - and he is probably the one driving this. If it is just a couple then crewing as a couple has value. However this is a family, most of whom have no idea about sailing, let a lone living on a boat.
That is the value of chartering, as I said in my original response. Its purpose is to introduce the family to the idea - and at the same time be a holiday. as said if they don't like that, then the project is doomed anyway.
The OP does not say what age the children are, and while it might be easier to take pre school children on such an adventure, once they start school they also start building their own lives which makes it much more difficult to take them away. The fact that the OP (where is he I wonder?) is inevitably thinking long term suggests his children will not be pre school when they start.
Personally I think the fixation on the "right boat" is wrong. Rarely do such projects fail because the boat was wrong. it is almost always crew related issues that lead to the breakdown. so before investing in the boat and making a live changing decision, best to concentrate on trying to ensure all the family buys into it. Difficult enough when there are only two people involved, but committing your children to an unconventional life is very high risk.
Sound advice but I'm not sure whether a fortnight drifting around the Ionian is preparation for living aboard or a catalyst for bitter recriminations when the electrics go down mid-Atlantic.