OK here's the drastic answer. I had a new house several years ago with many of the same issues, tried and tried for three years to get the crap put down by the builder to grow and even with reseeding, putting up all kinds of other methods tried below still not what I wanted.
I finally did this (and you'd better have a couple weekends and the patience and back to complete). Kill everything in the lawn with a general purpose weed/grass killer, everything. Wait the week or two to let it all turn brown then rake out as much crap as you can. Take a soil sample to the local extention office and they'll do a soil sample for under $10. Follow their advice and put lime (probably since you have clay) and some low dose (10-10-10) or similiar fertilizer down. Then get a tiller and till it all in 6-inches deep. Rake out and level, put your seed down and keep it damp. My neighbors admitted chuckling through their windows in the process but what came up was the thickest crap you could ever imagine. Not a weed or bit of crabgrass and it's been that way for years.
But again, it's a hell of a lot of work.
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