Even bermuda might be more effort than you want to put into this lawn. Here are some things to consider.

Check with your county ag extension agent (ask at the county court house) to see what kinds of prairie grasses do well in your area. There are many varieties of wheatgrass and a few will do will in your area. Ask specifically about those but also blue grama. If you really don't care what's growing, you could add some strawberry or Dutch white clover to the mix. Those grasses are not considered to be turf grasses, because they send up very tall seed stalks, but if you mow it once every few weeks, you likely won't see them. Generally these will go from rainstorm to rainstorm without water - except in extreme drought. These are bunch grasses which means they only become dense with more and more plants. Keep that in mind when seeding. They don't spread like bermuda, St Augustine, centipede, zoysia, or Kentucky bluegrass.

But before you do that, since you have a clean slate, now is the time to make sure your grading is perfect. You said the back yard sloped, but did not mention whether it sloped toward the house or away. The soil MUST slope away for at least 10 feet away from all buildings. At 10 feet the soil should be 6 inches lower than the grade at the house. The grade at the house should be 4 inches below the sill plate on the house. That makes a total of a 10-inch fall from the sill plate at 10 feet away. This should keep rainfall out of your basement. A landscaper would be able to do that for you. Whether you need to move your topsoil around is a landscaper decision, and it would be based on the basic drainage grading. The tool for finish grading is called a box blade and it goes on the back of a tractor, not a skidsteer or a Bobcat. An experienced finish landscape grader should be able to do the area around your house in a morning.

Do not rototill anything into anything. Work from the surface. Rototilling is a mistake in every case for a lawn. It creates an uneven (bumpy) surface almost immediately that gets worse for the next few years.

Your grading might be perfect already. In that case you need to sprout all the weeds you can before seeding. Otherwise the weed seeds will germinate along with the grass. You may as well start out with something nice. Start by watering lightly, 3x per day, every day for a week. At the end of the week spray everything with RoundUp. Continue watering for another week to get the slow sprouting weeds. Then spray again. Now it's time to seed. Scatter the seed and roll it down with a water fillable roller. Continue watering 3x per day until you get 80% germination of the grasses you planted. Then start to back off on watering.

As for your clay soil. Most people that think they have clay soil actually have soil with an imbalance of salts like calcium, potassium, magnesium, and sodium. Those can be fixed, but you have to know exactly what is wrong first. For that you need a really good soil test. The really good soil test lab is Logan Labs in Ohio. Your county agent might offer to have the soil tested, but if you want a much better test, spend $25 with Logan Labs and post the results on this forum. Check out some of the topics about soil tests to see what I'm talking about.

You can do a general test of your soil yourself to identify the components of sand, loam, and clay. Get a jar with straight sides. Fill it half way with dry soil and take a picture of it next to a ruler. Fill the jar all the way with water and a few drops of liquid soap. Shake it up and set it down. After 2 minutes take another picture with the ruler. This picture will show you how much sand you have. After 2 hours take another picture. This one will show you how much loam you have. After 2 days take another picture. If you cannot see through the water then you do have some clay. Clay particles will remain suspended for many days. If you can see through the water you likely don't have any clay.

Thanks for the thorough response. The slope is going away from the house, but I do have a landscaper who is going to shoot the elevations and make sure it is sloping enough as you describe. He will do some grading to correct, if necessary.

I guess regardless of whether it is red clay or red sand, the ground is basically hard as concrete now. My concern is if I replace the top soil that was set aside before construction on top that hard surface, that 1) heavy rain might just wash it away or 2) the grass roots will just grow into the few inches of top soil and never penetrate deeper.

I know you said tilling is a bad idea, but how would you address the two concerns I have if I don't loosen the clay/sand a bit and integrate into the top soil?

Thanks again

Soil test to determine clay content and salt balances. Both of these issues can be addressed by amendments on the surface.

Grow something. Anything. Roots in the ground are the best thing you can use to separate the soil particles and get a soil structure working for you. Put something down that will survive all winter and let it get tall. Taller grasses and plants will develop deeper roots. That's what you want.

Spray the area with shampoo. The rate is 3 ounces per 1,000 square feet. Follow that with 1/2 to 1 inch of irrigation or rain. The shampoo will allow the water to penetrate deeper into the soil where the temps are more moderate. The magical soil microbes will take over from there. They also can soften your soil as they populate your soil.

If you test and fix this soil that's fine. If you replace some of it with the topsoil, you will have to test and fix that. It all has to be fixed.