Interesting thread.
I think it depends on your child too.
2 of my dc are twins pretty similar academically imho as an ex teacher. Twin 2 is a total swot the other a lazy arse. The swot pushes himself(and me) nags to do his homework. Twin 1 I have to chase round the house to find him to do said homework.
Now I DO feel Twin 1 needs to be pushed more at school as he would happily get away with doing buggar all,he's very clever at pulling the wool over teacher's eyes. I find he's less interested when bored so it's a spiral effect.
Soooo I have crossed over to the dark side and could officially be classed as a pushy parent. Do I care? No not a jot as I'm not prepared to see him waste a perfectly good brain.
We don't do music lessons,tutoring etc though. I also dread any of my dc having any kind of role in school plays the possibility (after the tongue pulling debacle) of deep humiliation for myself always being at the back of my mind.