I am not a CSS expert, but I would think there must be a reason frameworks like Bootstrap went with the gutter approach? I don't know what it is, but I suspect there is a reason because otherwise this is so much simpler to deal with.

I think it's for consistency. If any container defines its own padding inside a column, the grid will quickly turn into a mess.

Yep! This is the main reason I use a hand-rolled grid and not one of the many common/popular ones.