"discing in" rye

Rye would almost grow on a glass bottle. It's been known to grow on the back of a pickup bed. I often just broadcast some into standing crops in late summer and it grows. I think your plan sounds great. I've done similar plantings like that in the past and they work well. Just don't plant rye in the spring, it doesn't do well because it's a fall planted crop. Plant oats if you want to plant in the spring.
Do you have a cultipacker? I sprayed, seeded, then cultipacked and have excellent growth with rye and clover i planted 2 weeks ago. Weve had good rains since planting though. Most of the seed was still laying on top of the dirt and would probably have failed if it germinated then turned off dry for a long time.
I will take a picture of what I am growing rye grain on that I threw on top of the bare ground tonight...last 4 years...
I also go for seed having contact with soil but never cover rye. You could mow, lightly disc and throw and do fine. Add some fertilizer
When I plant oats wheat and rye I process the oats wheat first, then broadcast rye on top. Same with clover.
It’s all in the timing of rain...no rain, no plot...

This is normally the week that I start planting, but it's just too dry and I have zero faith in the weather liars. I'm waiting until I can see a pattern of cold fronts. If that's mid October then so be it.
we mowed our rye 2 weeks ago. Have red clover under a lot of it but also hope for some fresh rye to resprout. We don't like to mow it when its wet because we have too much germination with all those rye heads. if things get a little grow happy we'll mow again in mid sept to keep that new rye at bay.
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.
I try to do my oats & wheat 1st,then the rye last as well. I understand that rain is important, but I believe also that you have to go pretty heavy on the rye.. at least for a couple years to get some OM built.

First year on my main 1acre I did around 250lbs then came back & top sown another 100lbs or so.

I’ll be doing that as well on the edges of the area we just cleared.

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Lightly discing to cover the seed works just fine. Did that for years before I purchased my tiller.
My opinion...if you just have 6-8" sparse vegetation on poor soil you need to do one of two things. Either find a better spot for your plot, or focus on getting that soil amended and worked to a point you can grow a successful plot before worrying about throw and mow. If it were me, I would spend the money on whatever amendments are recommended from the soil test, thoroughly incorporate those amendments the way the soil test intends and then seed the cereal rye using the best technique and equipment you have available. All seeds need soil contact and having rye covered with 1/2 - 1" of soil will help it germinate and reach moisture deep enough in the soil to withstand some adverse conditions.

You have to get over the initial hump before you can be in a place where something like throw and mow will work, and you quite possibly will have to turn some dirt to get there.