I am sorry that I did not tell you that I live in Northwest Ohio and yes Cool Season Grass .
I was told from a friend to seed a lawn than run the Spit Seeder over it in both directions leave the thatch on it and than put more seed over it and let the rain or water do the rest.
I feel that you should remove the thatch and put seed on the lawn.
Which way is the best way to do this? Yest it is a over seed, since we had a drought this Summer and had allot of dead areas.
You might ask way seed it than run the Spit Seeder over it without seed in it, the answer is the hopper will not hold seed any more.
If you have questions please ask and I will be happy to answer.