Doctor Who is a family show and was 'created' as a family show. The original lineup was made up of characters each targeting a different part of the family, susan - children, ian - teenagers, barbara - older women, doctor - older men. That's what it says on the production documents.
Secondly, the show has always been show in a 'family' slot, apart from the davison era. If it was a children's show, surely it would be in the same vicinity of sja in the schedules?
At the risk of patronizing you by telling you what a family show is - it contains childish parts for children and less childish parts for adults. It aims to have a foot in both camps because a typical nuclear family is half children half adults.
Some people complain it's too childish, some people complain it's unsuitable for 'exclusively' children. If it were a children's programme, these would be conflicting views. Since it is a family show, there is a balance.
In terms of recent pops at the show, it makes more sense to criticize a possible imbalance of immature and mature themes within the same show rather than to take half of the demographic out of the equation and then wonder why they are attracted to the show!
does that make sense? I might try and repost a clearer argument. sorry for the long post.