If your rye/thatch is that sparse, why even bother mowing. Spread your seed/fert then disc. We did it that way for years and our plots were awesome. Now that we have a cultipacker, they are even better. After discing, we throw our clover and chicory seed out, then cultipack over that.
We've tried throw and mow several times and it always fails or looks miserable compared to our other plots. Our deer will work on a new grain field pretty hard, so I will overseed some rye on top of the ground later in the season to thicken up the plot. It has mixed results. If the conditions are perfect, I will get some germination, but a lot of the seed just sits on top of the ground and does nothing.