They'll follow you around and do all those things with you if you do it. My six year old mows the lawn since he could push it. Each kids has a raised bed they tend to each summer. School is ok for socialization but a lot has to happen at home in terms of this agency thing.
My intuition tells me this is the crux of the matter: If I spend my time in the kitchen and the garden and wrenching and cutting and repairing, then my kid will be drawn to like activities. I hope I can behave accordingly and I hope you're right!