In our area, (Richmond VA), the "soil" is primarily clay. Builders just remove the clay from the house foudation / excavation work, then spread it around and grade it for sod. Many times the sod really does not take. Solution (long term) is to aerate like crazy in the fall and top dress / overseed / fertilize accordingly. So far I have had good results with the following sequence:
1. Spread gypsum / Lime over lawn.
2. Aerate aggressively in criss-cross pattern.
3. Overseed / fertilize
4. Diligent watering.
Some neighbors have then done top-dressing with high grade top soil, with excellent results, but it takes a lot of effort. I have seen some combine with aeration, others alternate aeration with top-dressing every other year.
In some smaller areas where I was more or less starting from scatch with seed, I would either aerator or used a garden rake, then top-dress with either Black Kow or peat moss, then seed, then more top-dress (lightly raked in), then dilgent watering.
As for removing cores. When I first moved here in '21 and saw how bad the clay was, I was real temped to rake them up and toss since it was all clay. Instead, I put down a ton of gypsum to help offset the clay. seemed to work for the most part. I have also gone back over freshly aerated / seeded lawn and spread / raked in peat moss / compost with good results.