Quite often I see middle aged men, cycling a bit quicker than pootling, older age teenagers hurtling along, people with no lights on at night deciding that it is unsafe to ride on the road so will use the pavement and nearly run you over because you can’t see them. I’ve seen whole families, mum father, two teenage children riding on the pavement, meaning pedestrians have to try and avoid them since they were riding two abreast. This lovely picture that you have tried to paint I haven’t witnessed.
When you are pulling a child, yes I can see why you would want to do that, no problem with that. Carrying? Do you mean in kid chair on the back of the bike? No problem with that. When escorting a child I would personally be on the road; each to their own.
I bet there are a lot of people with pushchairs, dog walkers, walkers who have a problem with a pushbike sharing their space. Yes it is their space since it is illegal to ride on a pavement; if they are not signed shared use then it is not ambiguous.