Oh, that's very m1469, indeed.

Can I simplify the answer? The general purpose of course is to get a well rounded education. There are only so much time in a lesson. If child does not practice much at home, there are no time left for choices in repertoire ( choices will be very limited) , theory, history and such. All focus goes into developing the basic skill which after all will not be developed, if child does not practice at home. Broader benefits in context of not practicing child? Phlease... You might as well send him to a play ground. He'll benefit just as much by interacting with other kids.
Don't take me wrong. You've asked a good question without specifying what is developed skill at playing an instrument, well rounded educations and fundamental parts means to you. Piano does not differ in any meaningful way from any other instrument in regard to your question. Taking your child to a concerts, listening to music at home, home performances and self study brings broader education. Sending a kid to a 30 minutes piano lessons once per week will not get you far, assuming child is practicing.
If you as a parent think there is only so much time in the day. I, as a teacher, think that there is only so much time in my lesson. In half an hour most children barely have time to play through a piece on the lesson ( ask your child to play a piece for you and time her ). I have to find the time to teach them something. There are as many philosophies out there as there are people it seems, but one seems to be constant - parents have little time, while teacher has enough, to give a child well rounded education in 10 on so minutes left after a child played his piece.
Private piano instructor is only a guide to your child's music education. I used to refuse 30 minutes lessons not that long ago, but at some point I figured - eh, it's your money after all.