ok this is coming from a kid who grew up in a landscaping family. if it were my father or grandfather doing the grass they would scrape 6 inches of final grade off and replace it with topsoil. (a mix of riverbed/ swamp drudge with mushroom mulch mixed in is what they always used and I have also only for my yard). reason being is what grass needs most is root growth. without thick roots grass burns easier and dies in the heat easier. with a deep root system it can hold water and moisture better and also tolerate drier hot times. i can prove this right in my parents yard. there is a section that i know the topsoil isn't as deep and there is more clay hardpan there. grass always burns first in the heat and requires more water. i also went through the same thing this summer with my neighbors french drain. it was topped with just hardpan dirt nothing grew or would burn up. we removed it and added 4 inches of good topsoil. green as could be now. a good topsoil will also hold moisture way more than a cheaper topsoil or clay. my yard and my parents yard will still be moist multiple days later than a yard without topsoil.
will it cost more, yes, but you will have a stronger lawn. whether you do seed or sod it applies. sod just gets u there quicker.
you also have to remember the type seed you are planting. ya if you plant fescue tap or short or red, whatever it will grow easier in any kinda dirt vs blue grass. blue is more delicate and can't take as much heat and dry dirt.
