So after checking the relevant chapter (32) I think Chinatsu's line about Winter's work actually refers to the root that Summer dug out.
Since presumably planting that was Winter's work (to bring forth Winter) and Summer digging them up is just preventing them causing another Winter and not actually bringing Summer.
So
We don't know how Spring/Summer/Autumn bring their season about.
But we knows of Winter (Winter's root, which these seeds might be theirs so Autumn prepare the seeds, Winter plant them, and Summer clean up the grown root later)