Your plan sounds good to me.
When we lived in the big city we had a load of sandy loam delivered in the spring. We never tilled or did anything. The grass came right up through the loam.
It might help to make a natural fertilizer with Epsom salts and molasses.
One part Epsom salts to two parts molasses. When dissolved aff to 5 gallons of water.
Do you compost? If not that is something to consider doing so you can have the benefits of making compost tea.
This thread might be of interest to you or others:
https://permies.com/t/11/organic-lawn-care-cheap-lazy