Mechanical aeration and twice-annual applications of compost at 1/2" deep works well for lawn health. Good dirt = good results.
The second is irrigation. Controlled irrigation over a lawn with excellent tilth pretty well takes care of all other potential problems.
I'd use chemicals the first year until the above is done. Properly. Some properties are quite far gone (going back decades). Starting over is never cheap, and that is the attraction of chemicals to act as substitute for mechanical labor.
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