Pure clay, huh? Have you done a jar test or another soil test showing the CEC (cation exchange capacity)? An overabundance of magnesium in a silty soil will mimic clay pretty well. For that matter an overabundance of magnesium in a sand soil mimics many of the features of clay. Fixing clay is different from fixing magnesium, so it is important to know what you have.

If you have clay, a really good way to modify it is to grow whatever grows as a cover crop to get lots of roots into the ground. Then kill off the cover crop and grow grass. The roots in the soil create a structural component that gets destroyed when you rototill. The roots also become the organic matter that you were thinking you could rototill into the soil with compost.

Is there any shade in the area you want this grass to grow?

Where do you live? Is this house really new to you?