Personally, I disagree that sooner is necessarily better. I think cultural expectations need to be taken into account, and in the UK, people channel their grief through very personalised funerals. Favourite flowers, favourite hymns, orders of service, readings from multiple family members. (Not saying that's wrong- that's the route my family went down!) But the thing is, none of that can be organised in under three days if it was an unanticipated death.
Yes, I agree, and maybe that's why I find the idea of which a quick funeral so culturally difficult. I think the timing of the funeral also maybe dictates cultural attitudes to it in other ways. I think that some other cultures see the funeral as the start of mourning, and that's why some people see it as so awful to delay it.
In my experience, by contrast, a funeral happens three weeks after a death, and in that culture the funeral marks not the start but a different stage of mourning - they're reflective, and so I can't imagine doing it three days after a death, when shock is still the dominant emotion. In this kind of culture they're a form of closure, so it makes sense that they come at a later stage.